Applied Science, Faculty of
Chemical and Biological Engineering, Department of
DSpace
UBCV
Tian, Hui
Doctor of Philosophy - PhD
University of British Columbia
Model predictive control (MPC) has attracted considerable research efforts and has been widely applied in various industrial processes. This thesis aims at developing economic MPC (econ MPC) strategies to optimize and control the nonlinear mechanical pulping (MP) process with two high consistency (HC) refiners, which is one of the most energy intensive processes in the pulp and paper industry. It possesses substantial economic motives and environmental benefits to develop advanced control techniques to reduce the energy consumption of MP processes. We propose four econ MPC schemes for nonlinear MP processes. Firstly, assuming that all the state variables are directly measurable, two different econ MPC schemes are proposed by adding different penalties on the state and input to ensure the closed-loop stability and convergence. Secondly, to address the issue of state variable off-sets from the steady-state target induced by above schemes, we further propose a multi-objective economic MPC (m-econ MPC) strategy. An auxiliary MPC controller and a stabilizing constraint are incorporated into the econ MPC. The stability of econ MPC is then achieved by preserving the inherent stability of the auxiliary MPC controller. Thirdly, to remove the assumption that all state variables are measurable, a moving horizon estimator (MHE) is employed to estimate the unmeasurable states. We then propose a practical framework integrating the m-econ MPC and MHE. Finally, we develop a tractable approximation for stochastic MPC (SMPC) to handle uncertainties associated with state variables. It can largely reduce the conservativeness or numerical instability incurred in robust or chance constraints of the traditional SMPC. The effectiveness of the proposed algorithms is validated by simulation examples of a nonlinear MP process consisting of a primary and a secondary HC refiner. It is shown that the proposed m-econ MPC schemes can significantly reduce the energy consumption (approximately 10\%-27\%) and guarantee the closed-loop stability and convergence. Therefore, the proposed methodology presents a great promise on practically implementing m-econ MPC to save costs for MP processes.
Stochastic Multi-objective Economic ModelPredictive Control of Two-stage HighConsistency Mechanical Pulping ProcessesbyHui TianB.Sc., Harbin Institute of Technology, 2011M.Sc., Harbin Institute of Technology, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Chemical and Biological Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)March 2020c© Hui Tian 2020The following individuals certify that they have read, and recommend to the Faculty of Graduate and Post-doctoral Studies for acceptance, the dissertation entitled:Stochastic multi-objective economic model predictive control of two-stage high consistency mechanicalpulping processessubmitted by Hui Tian in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin Chemical and Biological EngineeringExamining Committee:Bhushan Gopaluni, Chemical and Biological Engineering, The University of British ColumbiaSupervisorVictor Zavala, Chemical and Biological Engineering, The University of Wisconsin-MadisonSupervisory Committee MemberRyozo Nagamune, Mechanical Engineering, The University of British ColumbiaSupervisory Committee MemberMark Martinez, Chemical and Biological Engineering, The University of British ColumbiaSupervisory Committee MemberMatthew Yedlin, Electrical and Computer Engineer, The University of British ColumbiaUniversity ExaminerTsung Yuan (Tony) Yang, Department of Civil Engineering, The University of British ColumbiaUniversity ExamineriiAbstractModel predictive control (MPC) has attracted considerable research efforts and has been widely applied invarious industrial processes. This thesis aims at developing economic MPC (econ MPC) strategies to opti-mize and control the nonlinear mechanical pulping (MP) process with two high consistency (HC) refiners,which is one of the most energy intensive processes in the pulp and paper industry. It possesses substantialeconomic motives and environmental benefits to develop advanced control techniques to reduce the energyconsumption of MP processes.We propose four econ MPC schemes for nonlinear MP processes. Firstly, assuming that all the statevariables are directly measurable, two different econ MPC schemes are proposed by adding different penal-ties on the state and input to ensure the closed-loop stability and convergence. Secondly, to address theissue of state variable off-sets from the steady-state target induced by above schemes, we further proposea multi-objective economic MPC (m-econ MPC) strategy. An auxiliary MPC controller and a stabilizingconstraint are incorporated into the econ MPC. The stability of econ MPC is then achieved by preserving theinherent stability of the auxiliary MPC controller. Thirdly, to remove the assumption that all state variablesare measurable, a moving horizon estimator (MHE) is employed to estimate the unmeasurable states. Wethen propose a practical framework integrating the m-econ MPC and MHE. Finally, we develop a tractableapproximation for stochastic MPC (SMPC) to handle uncertainties associated with state variables. It canlargely reduce the conservativeness or numerical instability incurred in robust or chance constraints of thetraditional SMPC.The effectiveness of the proposed algorithms is validated by simulation examples of a nonlinear MP pro-cess consisting of a primary and a secondary HC refiner. It is shown that the proposed m-econ MPC schemescan significantly reduce the energy consumption (approximately 10%-27%) and guarantee the closed-loopstability and convergence. Therefore, the proposed methodology presents a great promise on practicallyimplementing m-econ MPC to save costs for MP processes.iiiLay SummaryMechanical pulping (MP) process is one of the most energy intensive processes in the pulp and paper indus-try. In order to reduce the energy consumption while improving final pulp properties, we develop advancedcontrol and estimation techniques for the two-stage high consistency (HC) MP process. With the proposedcontrol techniques, significant energy reduction can be achieved (approximately 10%-27%). Due to a lackof fast and reliable online measurement sensors in pulp mills, we also design a state estimator to estimatethe unmeasurable state variables from the limited measurement data. Simulation examples from a two-stageHC MP process are employed proposed optimization control techniques to demonstrate that significant im-provements in economic performance and pulp quality properties are achievable.ivPrefaceResults in Chapter 4 are based on the published paper:1. Hui Tian, Qiugang. Lu, R. Bhushan Gopaluni, Victor M. Zavala, and James A. Olson. Economic non-linear model predictive control for mechanical pulping processes. In Proceeding of the 2016 AmericanControl Conference, Boston, MA, USA, pp. 1796-1801, July 2016.The literature review, theoretical verifications, and simulations were based on my original idea and jointdiscussions with my supervisors Prof. Bhushan Gopaluni from the University of British Columbia (UBC),and Prof. Victor Zavala from the University of Wisconsin-Madison. Prof. Gopaluni, Prof. Zavala, Prof.Olson and Dr. Lu from UBC gave useful feedback during the research study and kindly reviewed the paper.Chapter 5 is based on the published paper:2. Hui Tian, Qiugang. Lu, R. Bhushan Gopaluni, and Victor M. Zavala. Multi-objective economicMPC of mechanical pulping processes. In Proceedings of the 55th IEEE Conference on Decision andControl, Las Vegas, NV, USA, pp. 4040-4045, December 2016.The idea of using multi-objective economic MPC for the control and optimization of a MP process was froma discussion between myself, Prof. Zavala, and Prof. Gopaluni. I performed the literature review, algorithmsanalysis, theoretical verifications, simulations, and paper writing for this research work. Prof. Gopaluni,Prof. Zavala, Prof. Olson, and Dr. Lu gave useful feedback during the research study and kindly reviewedthe paper.Chapter 6 is based on the published paper:3. Hui Tian, Qiugang. Lu, R. Bhushan Gopaluni, Victor M. Zavala, and James A. Olson. An economicmodel predictive control framework for mechanical pulping processes. Control Engineering Practice,vol. 85, pp. 100-109, 2019.vPrefaceI performed the literature review, algorithms analysis, theoretical verifications, simulations, and paper writingfor this research work. Prof. Gopaluni, Prof. Zavala, Prof. Olson, and Dr. Lu gave useful feedback duringthe research study and kindly reviewed the paper.The content of Chapter 7 has been submitted for journal publication and is under review:4. Hui Tian, Jagadeesan Prakash, Victor M. Zavala, James A. Olson, and R. Bhushan Gopaluni. Atractable approximation for stochastic MPC and application to mechanical pulping processes, submit-ted, 2020.This work develops a tractable approximation for stochastic MPC (SMPC). Under the proposed approach,we solve multiple deterministic MPC problems over individual scenarios of the uncertain variables to obtaina set of control policies and select from this candidate set a control input that yields the best approximation ofthe SMPC solution (i.e. yields the smallest statistical measure of the objective function (e.g., expected value)and of the constraints). The method was initialized from a research discussion between Prof. Prakash, atthe Department of Instrumentation Engineering, Madras Institute of Technology Campus, Anna University,Prof. Gopaluni, and myself. Prof. Zavala revised the detailed methodologies and made a rigid mathematicproof. I performed the literature review, algorithms analysis, theoretical verifications, simulations, and paperwriting. Prof. Zavala, Prof. Prakash, Prof. Olson, and Prof. Gopaluni gave useful feedback during this studyand kindly reviewed the paper.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAbbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Mechanical Pulping Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Current Issues and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7viiTable of Contents1.4.1 Adaptive/Dual Adaptive Control of MP Processes . . . . . . . . . . . . . . . . . . 71.4.2 Robust LQ Control of MP Processes . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.3 MPC of MP Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.4 Moving Horizon Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Mechanical Pulping Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Key Process Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 Main manipulated variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Main Process State/Operating Variables . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Pulp Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.4 Process disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.5 Unmeasurable state variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Two-stage HC MP Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Nonlinear MP Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 General Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Pulp Quality Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Traditional Tracking MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Economic MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.1 Economic MPC (Econ MPC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 Multi-objective Economic MPC (M-econ MPC) . . . . . . . . . . . . . . . . . . . 253.3.3 Stochastic MPC (SMPC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Nonlinear MPC (NMPC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Stability of NMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5.1 Nominal Stability of NMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32viiiTable of Contents3.5.2 Robust Stability of NMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6 IPOPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Econ MPC for a Two-stage HC MP Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Econ MPC for the MP process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.1 Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.2 Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.1 Simulation I: Comparison of Econ MPC with Different Penalty Methods . . . . . . 434.3.2 Simulation II: Energy Reduction by Using Econ MPC Compared with StandardMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Multi-objective Economic MPC for the MP Process . . . . . . . . . . . . . . . . . . . . . . . 505.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.2 Multi-objective Control Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.2 Stability of M-econ MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3 M-econ MPC Design for a Two-stage HC MP Process . . . . . . . . . . . . . . . . . . . . 535.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 MHE and the Integration of MHE and M-econ MPC . . . . . . . . . . . . . . . . . . . . . . 616.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 State Estimator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.2.1 Full Information Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.2.2 Moving Horizon Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.3 MHE Design for a Two-stage HC MP Process . . . . . . . . . . . . . . . . . . . . . . . . . 656.4 Integration of MHE and M-econ MPC for MP Processes . . . . . . . . . . . . . . . . . . . 66ixTable of Contents6.4.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.4.2 MHE and M-econ MPC Framework for MP Processes . . . . . . . . . . . . . . . . 686.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 A Tractable Approximation for Stochastic Model Predictive Control and Application to MPProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2 Stochastic Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.3 Approximating the Policy of SMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.4 Application to Mechanical Pulping Processes . . . . . . . . . . . . . . . . . . . . . . . . . 827.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92xList of Tables2.1 A list of process variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Basic steps of MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.1 The implementation of m-econ MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 The variations of chip bulk density dc and chip solid content sc of the raw material from theirnominal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.1 The implementation of m-econ MPC and MHE for a two-stage HC MP process . . . . . . . 707.1 Implementation of proposed approximate SMPC scheme in ideal case . . . . . . . . . . . . 817.2 Implementation of proposed approximate SMPC scheme in general case . . . . . . . . . . . 877.3 Simulation parameters for the SMPC controller . . . . . . . . . . . . . . . . . . . . . . . . 877.4 Variances for outputs, inputs, and SE for scenario cases S = 1,5,30 . . . . . . . . . . . . . . 88xiList of Figures1.1 Operational units in a typical MP process . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Multilayer automation decision hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Graphical depiction of the integrated controller and estimator for the MP process . . . . . . 62.1 Schematic of two-stage HC MP process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Basic concept of MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1 Block diagram for model predictive control of a typical MP process . . . . . . . . . . . . . 414.2 Pulp quality after two-stage HC refining by using econ MPC in simulation 4.3.1 . . . . . . . 444.3 The state variables of the MP process by using econ MPC in Simulation I . . . . . . . . . . 474.4 The manipulated variables of the MP process by using econ MPC in Simulation 4.3.1 . . . . 484.5 Comparison of the energy reduction in Simulation 4.3.1 & 4.3.2 . . . . . . . . . . . . . . . 495.1 Pulp qualities after two-stage HC refining . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 The state variables of the MP process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 The manipulated variables of the MP process . . . . . . . . . . . . . . . . . . . . . . . . . 595.4 Comparison of the energy reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.1 Integrated controller and estimator for MP process . . . . . . . . . . . . . . . . . . . . . . 686.2 The manipulated variables of the MP process . . . . . . . . . . . . . . . . . . . . . . . . . 726.3 The state variables of the MP process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.4 SE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.1 Closed-loop output policies obtained with the proposed controller . . . . . . . . . . . . . . 847.2 Closed-loop input policies obtained with the proposed controller . . . . . . . . . . . . . . . 857.3 Comparison of the energy reduction achieved with the proposed controller . . . . . . . . . . 86xiiSymbols∈ belongs to⊆ subset of∑ summation||x|| norm of a vector x||x||2Q weighted norm xT QxQ> 0 a positive definite matrix Qmin minimummax maximum∀ for all6= not equalR≥0 non-negative real numbersRn n-dimensional Euclidean spacef : Rn×Rm 7→ Rp function f maps Rn×Rm into a space Rp/0 empty setdiag(w) a diagonal matrix with w vector in the diagonallim limit∞ infinity→ tend to∃ exists+∞ positive infinityxiiiAbbreviationsAMPL A mathematical programming languageCSF Canadian standard freenessCTMP Chemical-thermo-mechanical pulpingDMPC Deterministic MPCEcon MPC Economic MPCGPC Generalized predictive controlHC High consistencyi.i.d Independent and identically distributedIPOPT Interior Point optimizationISS Input to state stabilityKKT Karush-Kuhn-TuckerLC Low consistencyLFC Long fibre contentLQ Linear-quadraticM-econ MPC Multi-objective economic MPCMHE Moving horizon estimatorMIMO Multi-input-multi-outputMP Mechanical pulpingMPC Model predictive controlNLP Nonlinear programmingNMPC Nonlinear MPCPQM Pulp quality monitorRI Refining intensityRTO Real-time optimizerxivAbbreviationsSC Shive contentSE Specific energySMPC Stochastic MPCSRP Specific refining powerSSMPC Scenario-based stochastic MPCs.t. Subject toTMP Thermo-mechanical pulpingTr MPC Tracking MPCTSE Total specific energyw.r.t With respect toxvAcknowledgmentsFirst, I would like to express my sincere gratitude to my Ph.D. supervisor, Professor Bhushan Gopaluniat the Department of Chemical and Biological Engineering, UBC. During my Ph.D. life, I have had thegreatest honor and pleasure of working with him. He has been always very supportive and encouragingto his students. He introduced to me this industry-oriented project and provided me an excellent researchenvironment and guidance towards solving practical industrial problems.I am highly grateful to Professor Victor Zavala at the Department of Chemical and Biological Engineer-ing, the University of Wisconsin-Madison, for all his generous help through developing core theoretical ideasto the final paper writing. His expertise and knowledge in system optimization and model predictive con-trol techniques are invaluable to our research work. Completing this project would have been more difficultwithout his guidance.My sincere appreciation also goes to Professor James Olson, the Director of Pulp and Paper Center(PPC), UBC, for providing the financial support, valuable guidance, and discussions in this project. Mygreatest thanks extend to Professor Jagadeesan Prakash, at the Department of Instrumentation Engineering,Madras Institute of Technology Campus, Anna University, India, for his help in our work of developing thestochastic MPC during his visit to UBC. I would also like to thank my project manager Meaghan Miller.Meaghan gave me numerous assistances in realizing our algorithms in the pulp mill. I appreciate thoseproductive bi-weekly project meetings organized by Meaghan. I also acknowledge her for arranging multiplepulp mill trips. These trips and visits are of vital importance for the success of this project.I would also like to thank all the collaborators and colleagues in my Ph.D. project and study, DavidZamar, Lee Rippon, and Yiting Tsai.I am gratefully acknowledge the financial contributions from the Natural Sciences and Engineering Re-search Council of Canada through the Collaborative Research and Development program and the support ofour project partners: AB Enzymes, Alberta Newsprint Company, Andritz, BC Hydro, Canfor, Catalyst Paper,FPInnovations, Holmen Paper, Millar Western, NORPAC, Paper Excellence, Quesnel River Pulp, Slave LakexviAcknowledgmentsPulp, Westcan Engineering, and Winstone Pulp International. Our special thanks goes to Alberta Newsprint,FPInnovations, and Howe Sound Pulp and Paper (Paper Excellence) for providing data for our models.Finally, I must express my gratitude to my parents and my family. They have always been a constantsource of motivation for me. They have been always standing by me during the most difficult time of myPh.D. life. Their patience, inspiration, and infinite love are always priceless to me. This thesis would not bepossible without them. I would also like to thank my little baby, Mason, who has brought me so much joyduring the last year of my Ph.D life.xviiDedicationTo My Family.xviiiChapter 1Introduction1.1 General IntroductionA wood pulping process converts wood chips into paper-making fibres. The mechanical pulping (MP) pro-cess, which includes thermo-mechanical pulping (TMP), chemical-thermo-mechanical pulping (CTMP), andbleached chemical-thermo-mechanical pulping, is a vital but significantly energy-intensive operation in thepulp and paper industry [4, 23, 29, 32, 35, 62, 66, 78]. Conventional MP process has two high consistency(HC) refiners – the primary and the secondary HC refiner. In the pulp and paper industry, the consistency isdefined as the mass ratio of the dry fibre to the mixture of dry fibre and water. HC refers to the ratio between20% and 50% whereas the low consistency (LC) refers to the ratio between 3% and 5%. The HC refinersplay a dominant role in achieving the required pulp properties, yet consume nearly 60% of the total electricenergy consumption of the MP process [43].In recent years, the MP industry has been facing a number of new challenges. Firstly, the increasingelectrical energy prices nowadays make the economic controller design significantly demanding. About 1/3the cost of mechanical pulp is electrical energy, which has been rapidly increased in recent years. Thisincrease is driving the industry to develop new process and controller designs that make it significantly lowenergy. The second challenge is the changing of the demand markets and products in the past decade. Thedemands for newsprint and other printing and writing paper are rapidly declining at a rate of more than10% per year, while the demand for packaging and absorbent products continues to increase globally. Thisresults in MP mills moving into producing products that have dramatically different quality requirementsthan conventional products. Thus, in order to retain the market and customers, it is imperative for MP millsto develop advanced control techniques that can reduce electrical energy consumption and enforce strict pulpquality specifications to face competitors globally. Targeting on the two challenges, this thesis focuses ondeveloping advanced control and optimization strategies for MP processes to reduce the energy consumptionand improve pulp qualities.11.2. Mechanical Pulping ProcessThis chapter is outlined as follows. A brief introduction of the mechanical pulping process is given inSection 1.2. Section 1.3 is focused on the existing issues and challenges on the application of the advancedcontrol techniques to the MP processes. The motivation and objective of our thesis are also provided in thissection. The literature review is offered in Section 1.4. The contributions of this work are summarized inSection 1.5, followed by the outline of this thesis in Section 1.6.1.2 Mechanical Pulping ProcessA multi-stage MP process generally consists of wood chip pretreatment, wood chip refining, and pulp re-fining as shown in Figure 1.1, for which a brief introduction is provided as follows. Readers can refer to[29, 32, 43] for more detailed process descriptions.In the wood chip pretreatment stage, wood chips are screened to remove over and under sized particlesand then washed to remove contaminants, such as rocks and sand. The chips are then be steamed andpreheated at atmospheric pressure around 100◦C. For CTMP, some preliminary chemical pretreatments willalso be carried out at this stage to improve pulp properties such as brightness and strength.In the wood chip refining stage, there are typically two HC refiners, namely the primary and secondaryrefiner as shown in Figure 1.1 and 2.1. After the pretreatment in the first stage, the wood chips are introducedinto the inlet of the primary HC refiner by the cylindrical chip transfer screw feeder. Dilution water is usuallyfed into the inlet of the refiner to control the consistencies in the refining zone. The wood chips are brokendown into fibres as they pass through the two rotating discs of the refiners.As the final stage, the wood chip pulping unit, which works at LC, is designed to further enhance the pulpproperties, such as the brightness and strength. After the secondary HC refining, the wood pulp is fed to thelatency chest for latency removal and fibres straightening. In the third stage LC refining, the pulp propertieswill be further controlled to achieve the required final pulp property requirements. The pulp screening andreject refining decreases the shive content and remove unrefined fibre bundles. The wood pulp is then washedto remove wood resins and metallic ions. After pulp watering, the dewatering and pulp bleaching will beperformed to increase the brightness and improve optical properties.21.3. Current Issues and ChallengesFigure 1.1: Operational units in a typical MP process1.3 Current Issues and Challenges1.3.1 MotivationThe MP process is one of the most energy-intensive operations in the pulp and paper industry [4, 23, 29, 35,61, 62, 78]. During the last two decades, the MP process has been intensively studied in terms of refiningoptimization, energy reduction, and pulp quality improvement [3, 23, 29, 61, 61, 66, 78]. However, thedevelopment of advanced control strategies is difficult and relatively primitive [32].The two-stage HC MP process is an inherent multi-input-multi-output (MIMO) operation with severenonlinear interactions and couplings between variables. The MP process model can be developed using themechanistic or the empirical method. Most of the existing work on the advanced controller design for the MPprocess is based on linear empirical process models in the last two decades [3, 4, 34–36, 40]. Even thoughthe linear model is simple to implement, it approximates the process over a limited range of operations,which limits its application [32].As one of the widely used advanced control techniques in the pulp and paper industry, MPC is anoptimization-based advanced control technique that computes optimal control policies by solving a finitehorizon optimization problem in real time. An outstanding feature of MPC is that physical constraints onactuators and outputs can be incorporated directly in the optimization problem. As a result, MPC has at-tracted considerable research efforts and has been widely applied in various industrial processes [17, 79, 97].The main goal of advanced control is to operate the plant as close as possible to the economically opti-mal operating point such that the net return is maximized in the presence of disturbances and uncertainties,31.3. Current Issues and Challengeswhile ensuring stability. In the standard MPC implementation, the optimal steady-state operation is typi-cally addressed by a control hierarchy involving multiple layers as shown in Figure 1.2 [38]. Taking theinformation from the market and customer, the planer and scheduler focus on economic forecasts and thetiming of actions and events. The planing and scheduling unit, which works daily or monthly at the top ofthis hierarchical structure, provides the plant time schedule, the production plan, and parameters of the costfunction and constraints [6]. In the next layer, a real-time optimizer (RTO) computes the optimal operatingpoints by solving a static optimization problem based on a nonlinear steady-state plant model. The RTOemploys a stationary complex model of the plant and works on an hourly or daily basis. It implements theeconomic decision in real time and provides setpoints to the advanced control layer, namely the MPC layerwhich is designed using a linear dynamical model.Relevant results on the control of MP processes using the multi-layer control structure with a linear MPChave been studied in the past decade [29, 32, 40]. However, since the economic setpoints calculated by RTOmay be inconsistent or unreachable to the advanced control layer, the economic performance of the MP plantis inadequate. Moreover, the traditional MPC is designed to ensure the asymptotic tracking to the setpoint.The transient costs, on the other hand, is neglected by the advanced control layer. Thus, this multi-layercontrol frame is only optimal when the setpoint does not change over time in the advanced control layer.However, in some MP processes, different types of pulps are produced in a day, and the correspondingsetpoints change frequently. In this case, the transient cost should be treated as important as the steady statecost. Hence, it is very important to optimize the cost of the entire trajectory, not only at the steady state [39].Moreover, due to the presence of inherent nonlinearities of the MP process and inconsistence in the modelsbetween the RTO layer and the advanced control layer, the linear model at the advanced control layer has tobe re-identified each time when the setpoint changes, which limits the implementation of MPC in MP mills.The majority of the existing work on advanced control for MP processes assume that all the state vari-ables and the pulp properties after each refiner are available [44–46]. However, due to a lack of fast andreliable measurements online sensors, not all the important variables are available, which prevents the im-plementation of advanced control techniques in pulp mills [32]. For instance, one of the most widely usedsensors in pulping industry is the Pulp quality monitor (PQM) which measures the shives, fibre size distri-bution, and freeness in the pulp. This sensor takes measurements every 50-60 minutes, which limits the useof MPC. Thus, the state estimator design is of vital importance for the pulping industry. State estimationfor nonlinear MP systems is particularly challenging, especially when there are constraints on state variables41.3. Current Issues and ChallengesFigure 1.2: Multilayer automation decision hierarchy[71, 72, 81]. To address such issues, moving horizon estimation has been proposed as a practical approachthat can directly embed nonlinear dynamics and constraints [1, 81].1.3.2 ObjectivesFor a two-stage HC MP process in the pulp mill, it is vital to reduce the energy consumption and the variancesof the state variables while guarantee the convergence and stability of the closed-loop MP process. This thesisfocuses on three aspects. Firstly, we build a nonlinear dynamic model for the two-stage HC MP process.Then we attempt to investigate economic MPC (econ MPC) frameworks based on this model, focusing onthe energy cost reduction while preserving the closed-loop stability. Third, state estimator is developed toestimate the unmeasurable states and integrated into the closed-loop MPC. The eventual control-estimationblock diagram for the closed-loop MP process is illustrated in Figure 1.3.1.3.3 Problem StatementThe two-stage HC MP process is one of the most energy intensive units in the pulp and paper industry. Savingelectricity energy and improving the pulp qualities in MP mills in the past two decade have become crucialdue to the increasing electricity price and strong global competitions. Therefore, it is desirable to developadvanced control and optimization techniques to decrease the energy consumption while improving the pulpproperties and ensuring the closed-loop stability. We will identify the nonlinear two-stage HC MP process51.3. Current Issues and ChallengesMechanical Pulping ProcessOutput Inputm-econ MPCOutput InputMHE𝑦𝑦𝑢𝑢 𝑢ො𝑥Figure 1.3: Graphical depiction of the integrated controller and estimator for the MP processmodel based on the collected industrial data and mechanical analysis. We will also propose different formsof nonlinear econ MPC and apply them to the two-stage HC MP process through simulation experiments.Closed-loop stability analysis will be performed with the proposed nonlinear econ MPC in the two-stage HCMP process. To estimate the unmeasurable states due to a lack of reliable and fast online sensors, we willdevelop a state estimator using the moving horizon estimation approach. The proposed nonlinear econ MPCcontrollers will combine the state estimator into an integrated estimator-based MPC scheme.Two-stage HC MP Process Model: Even though significant research has been in literature in terms ofrefining optimization, energy reduction, and pulp quality improvement, the application of advanced controland optimization strategies is still difficult and relatively primitive due to the unknown process mechanismand strong nonlinearity in the two-stage HC MP process. The two-stage HC MP process is an inherentlynonlinear MIMO process with strong couplings between process variables. To gain a deeper understandingof the unknown mechanism inside the two-stage HC refining process, in-depth nonlinear process modelswill be provided based on the industry data and mechanical analysis. Moreover, a balanced trade-off will bemade between the complexity and accuracy of the nonlinear process model and the potential computationalburden in solving the model based control and optimization problems.Nonlinear Econ MPC Design: We will propose different nonlinear econ MPC methods, which are ableto merge the traditional multi-layer hierarchy of the control structure, to decrease the energy consumption ofthe two-stage HC refining process while ensuring the closed-loop stability. A concrete and detailed stabilityanalysis of the closed-loop two-stage HC MP process will be present to verify the stability and effectivenessof the proposed control techniques. We will validate the proposed control and estimation methods throughextensive simulations.State Estimator Design: We will use the moving horizon estimation approach to estimate the unmea-61.4. Literature Reviewsurable state variables in the process. We will then provide a new framework to combine the state estimatorand the proposed advanced control techniques. The finial control and estimation framework is structured inFigure 1.3. This new framework will make it possible for the pulp mill to implement the proposed advancedcontrol technique where only limited online sensors are available.1.4 Literature ReviewThe control of mechanical pulping processes has been in a state of ongoing development since the mid-1970s [23, 29, 32, 35, 60–62, 90]. Intensive studies have been made by researchers and engineers towardsrefining optimization, energy reduction and pulp quality improvement [3, 4, 34–36, 40, 43–46, 95]. Broadlyspeaking, the current control techniques for the MP process can be categorized into the following threegroups: adaptive control, robust linear-quadratic (LQ) control, and MPC.1.4.1 Adaptive/Dual Adaptive Control of MP ProcessesIn the MP process, the control of motor load through manipulating the gap size is difficult because of thenonlinear relationship between the plate gap and the motor load [3, 4, 34, 36, 83]. The first attempt atapplying adaptive controller to the motor load control in a MP process dates back to the 1980s [83]. In[34], an adaptive control scheme was proposed, which consisted of a recursive least-square process modelestimator with a forgetting factor, and a Dahlin controller whose parameters were tuned in real-time basedon the updated model. Trials on industrial MP processes showed that the proposed regulator was capableof tracking slow gain drift and avoiding pulp pad collapses. This method was further improved in [36] inwhich the variable forgetting factor was replaced by fault-detection techniques. Based on the above results,a dual adaptive control strategy was proposed in [4] in which it took account of the process nonlinearity. Asuboptimal dual control strategy to a reject refiner was reported in [3]. However, the computational burdenof solving the aforementioned adaptive/dual adaptive control problem online restricted the applications ofthis technique in pulp mills [32]. The author in [54] developed an adaptive control strategy to control thefreeness of the pulp through manipulating the gap size. With this method, rapidly measured specific energy(SE) was required in order to infer the freeness based on a dynamic-noise model. However, this method wasa single-variable control approach which limited its implementations in practical applications [32].71.4. Literature Review1.4.2 Robust LQ Control of MP ProcessesLQ control is one of the most fundamental optimal control techniques in which the system dynamics aredescribed by a set of linear differential equations and the cost function in a quadratic form [12]. A minimaxrobust LQ control of freeness in a MP plant was proposed in [95] in which the wear and tear of the refinerplates was considered. The basic idea of the robust LQ control method in [95] was to identify the MPprocess models at different operating points and obtain a minimum output variance by solving a small LQproblem at each operating point. The final control move was calculated by minimizing the maximum ofthe output variance over the set of operating points which subject to a constraint on the maximum inputvariance. One disadvantage of this method was that the proposed LQ control law only provided satisfactoryperformance when the range of parameter changes was not too large. The authors in [98] presented a designprocedure of robust LQ controllers for the refining process. Three major topics were discussed in the refiningprocess including the mathematical modeling of the process with large uncertainty, a performance index foroptimizing the process, and the design of LQ control system with robust stability.1.4.3 MPC of MP ProcessesIn recent decades, MPC has become one of the most successful controllers which have a significant impact onthe advanced control of multi-variable processes [17, 79, 97]. It is an optimization-based control techniquethat computes optimal control steps by solving a receding horizon optimization problem at each samplingtime. One of the most important features of MPC is that physical constraints can be incorporated into theoptimization problem directly, which makes it one of the most active research topics and the most popularadvanced control techniques in the process industry [17, 79, 97].In the context of MP processes, implementations of linear MPC have been reported in the past decade[29, 32]. The constrained MPC technique for the wood chip refining process was first proposed in [33].In this paper, the controller was designed by using generalized predictive control (GPC) method with thehelp of quadratic programming. The linear MPC control techniques for a TMP process were proposed in[29] and [30]. The objective was to control the SE and consistency in a TMP process by manipulatingthe screw speed, the dilution flow rate, and the plate gap size. These implementations used a linear MPprocess model, which resulted in suboptimal performance due to the presence of strong nonlinearities. Thedevelopment of the advanced optimization algorithms and computing hardware enabled the direct handling81.5. Summary of Contributionsof the nonlinear dynamics in MPC formulations [37, 55, 58, 75]. The authors in [40, 44–46] presented resultson the control and optimization of MP processes using nonlinear MPC (NMPC). In [40], a non-constrainedpredictive control technique based on a nonlinear Laguerre model was developed for MP processes. Themore recent results on the control and optimization of MP processes using nonlinear econ MPC can befound in [43–46]. In [44], the authors proposed a general NMPC framework for setpoint tracking of a two-stage TMP process. The dynamics of the two-stage TMP were described by empirical and first-principlemodels. The computational burden of the associated nonlinear programming (NLP) problem was handledusing the advanced step NMPC concept with the Interior Point OPTimizer (IPOPT) solver [104]. Thiswork was further developed in [45], where a dynamic optimization scheme to simultaneously regulate andoptimize the MP process was proposed. For the multi-stage MP process, the first attempt of using the NMPCtechnique appeared in [46].1.4.4 Moving Horizon EstimationThe objective of state estimation is to reconstruct the evolution of the process state from the measurementsand the past input signal. Moving horizon estimator (MHE) uses online optimization for designing stateestimators. In MHE, the state estimate is determined online by solving a finite horizon state estimationproblem. At each sampling time, the old measurements are discarded from the estimation window whennew measurements are available. MHE is formulated as an optimization problem, which makes it possibleto handle nonlinear systems and inequality constraints on decision variables explicitly. The early applicationof MHE for linear system can be found in [91]. The application of MHE for a nonlinear process was firstlyreported in [48], where the state estimator was developed to estimate the initial state of the system while statedisturbances were completely ignored. Applications of MHE can also be found in other processes, such ashybrid systems [11], bioprocess system [80], batch crystallization process [56]. Conditions for the stabilityof state estimation can be found in [13, 73, 81, 82].1.5 Summary of ContributionsThe main contributions of this thesis are summarized as follows,1. We investigate a nonlinear tracking MPC (tr MPC) for a two-stage HC MP process. This part ofthe research work is acted as a comparison benchmark for the rest econ MPC design in the tracking91.5. Summary of Contributionsperformance as well as the energy reduction in terms of SE.2. We design a nonlinear econ MPC for a two-stage HC MP process. To guarantee the convergenceand stability of the system, two types of regularization terms are added to the nonlinear econ MPCformulation. The first type of regularization includes the L2-norm of the deviations of state variablesfrom their steady-state targets and the L2-norm of the increments between consecutive manipulatedinputs. The other type of regularization consists the L2-norms of the difference of the states andmanipulated variables from their respective steady-state setpoints. Control performance and energyreduction capability are compared between these two types of econ MPC. Simulations are conductedto show the effectiveness of the propose approaches. Particularly, the proposed controllers show goodtracking performance and energy reduction on the simulated two-stage HC MP process. Additionally,the proposed controllers outperforms the standard MPC in terms of the energy consumption.3. We propose a multi-objective economic MPC (m-econ MPC) for a two-stage HC MP process. Toguarantee the closed-loop stability of the m-econ MPC, an auxiliary tracking MPC is incorporatedin the m-econ MPC as a stabilizing constraint. Then the stability of the m-econ MPC is ensured bypreserving the inherent stability of the auxiliary tracking MPC. By adjusting the scaling parametersthe energy reduction is between 10%–27%. Since the stability of nonlinear MP process is still in itsprimitive stage, this research work will be a significant contribution to this field.4. We develop a moving horizon estimator (MHE) to estimate the unmeasurable state variables in theMP process. MHE is an optimization-based state estimator. It estimates the unmeasurable states andaddress the noises in the measurements by solving a fixed size and forward moving window optimiza-tion problem which minimizes the difference between the available measurements and the predictedoutputs. We also propose to integrate the m-econ MPC and MHE. This new control-estimation frame-work makes it possible to apply the m-econ MPC technique in pulp mills where only limited onlinesensors are available.5. We propose an approximation of stochastic MPC (SMPC) to determine the optimal control policy fornonlinear MP systems affected by random disturbances and/or uncertainty. The proposed algorithmdecomposes the original SMPC into a set of scenario-based optimization problems. The optimal inputis calculated respectively for each scenario. The best overall optimal input is then selected among101.6. Outline of This Thesisthese optimal inputs in each scenarios based on certain rules. An advantage of this method is that thesolutions obtained are statistically robust as they are not easily influenced by extreme scenarios whichhave a low probability of occurring. Another advantage of is that we can avoid the computationalchallenges associated with stochastic optimization by decomposing the problem into a set of similardeterministic optimizations which can be efficiently solved using parallel computation.1.6 Outline of This ThesisThis thesis covers control, optimization, and state estimation for a two-stage HC MP process.In Chapter 2, the detailed model of a two-stage HC MP process are proposed based on the data collectedfrom the industry. The state variables, the manipulated input variables, the pulp properties, and the distur-bance of the MP process are defined. In Chapter 3, the general MPC is introduced, including the problemset-up and stability analysis. Discussions on the solution of large-scale NMPC problems are also given inthis chapter. This chapter acts as a general foundation for the rest of the paper. In Chapter 4, two typesof econ MPCs with different penalization terms for the MP process are presented. In the first type of econMPC, the deviation of the states from their steady-state targets and the consecutive increment of manipu-lated variables are added in the objective function. In the second scheme, the deviation of both state andmanipulated variables from their steady-state setpoints are included in the cost function. Simulations areconducted to show the effectiveness of the propose approaches. In Chapter 5, the m-econ MPC is proposed,and detailed stability proof of the closed-loop m-econ MPC is provided. An auxiliary tracking MPC con-troller is incorporated into the econ MPC as a stabilizing constraint. Simulations on a two-stage HC MPprocess with different scaling parameters are used to show the effectiveness of the proposed m-econ MPC onthe tracking performance as well as the energy reduction rate. In Chapter 6, the MHE is proposed. A newcontrol-estimation framework, i.e. the integration of the proposed m-econ MPC and MHE, is also includedin this chapter. Simulations based on the m-econ MPC and MHE on a two-stage HC MP process illustratesthe effectiveness of the proposed controller and MHE. In Chapter 7, we propose an approximation stochas-tic MPC for the two-stage HC MP process. Case studies are presented to demonstrate economical benefitsand reduction in the total energy consumption as well as the variability in the state variables and the pulpproperties. Chapter 8 concludes of this thesis, and provides future work.11Chapter 2Mechanical Pulping Process Model2.1 IntroductionThe traditional two-stage HC MP process, as is depicted in Figure 1.1, is an inherently MIMO nonlinearsystem. The process modeling of MP process is still in its undergoing stage due to the unclear mechanisminside of the HC refiners [32, 43].In this chapter, we provide detailed models of the complex two-stage HC MP process. The key processvariables, including the main manipulated variables, process state variables, pulp properties, process dis-turbances, and the unmeasurable state variables, are defined in Section 2.2. Based on the defined processvariable, we build the nonlinear two-stage HC MP process model in Section 2.3. A general process modeldescription is also given in this section. The pulp quality modeling is introduced in Section 2.4. Section 2.5summarizes this chapter.2.2 Key Process VariablesThe key variables required for the control and optimization of MP refining processes, such as manipulatedvariables, operating variables, and pulp quality variables, are discussed in detail in this section.2.2.1 Main manipulated variablesManipulated variables are the input variables which can be adjusted during the process operation. The maininputs in a MP process, shown in Figure 2.1, are summarized in the left column of Table 2.1.(1) Chip transfer screw speed (rmp), u(1). The chip transfer screw speed is the main manipulated variableused to control the flow of chips from the preheater at the wood chip pretreatment stage to the inlet ofthe primary HC refiner. Any changes in the screw speed can affect the flow of dry fibres to the refiner.Most pulp mills use the transfer screw speed to set the desired production rate.122.2. Key Process VariablesConicalscrew feederSteam ventilation chamberand cylindrical screw feederPrimary refinerWood chipsPrimary motor𝑢(1)Secondary motorSecondary refinerPulp property after primary refiningPulp property after secondary refiningPulp after primary refinerSteam flow rateSteam flow rate𝑢(2)𝑢(3)𝑢(4)𝑢(5)𝑥(2)𝑥(1)𝑥(4)𝑥(3) 𝑥(5)Figure 2.1: Schematic of two-stage HC MP processTable 2.1: A list of process variablesMV Note Notation SV Note Notationu(1) Chip transfer screw speed (rpm) R x(1) Production rate (tonnes/day) Pu(2) Primary refiner plate gap (mm) Gp x(2) Primary motor load (MW) Mpu(3) Primary dilution flow rate (kg/s) Dp x(3) Primary consistency (%) Cpu(4) Secondary refiner plate gap (mm) Gs x(4) Secondary motor load (MW) Msu(5) Secondary dilution flow rate (kg/s) Ds x(5) Secondary consistency (%) Cs(2) Primary/secondary refiner plate gap (mm), u(2), u(4). The plate gap is the distance between two platesof a refiner. It is normally controlled by a mechanical loading system. The gap size can be measuredby a gap sensor or indicated by changes in the relative shaft position. Variations in gap size directlyimpact the mechanical force exerted by plates onto wood chips, and thus affect the motor load.(3) Primary/secondary dilution flow rate (kg/s), u(3), u(4). The refining zone consistency has a majoreffect on pulp properties. The water added to the refiner will alter the consistency and thus change thepulp quality. Large variations in the dilution water flow rate can lead to unstable refining operation.Manipulating the water flow rate at the inlet of each refiner is commonly used by pulp mills to maintainthe consistency in refiners.2.2.2 Main Process State/Operating VariablesState variables are the key variables and highly correlated with pulp properties. By controlling manipulatedvariables the states are maintained at optimum setpoints. The optimum setpoints in a MPC controller aredetermined by the economical operation of the MP process. Main state variables used in this thesis are132.2. Key Process Variablesshown in Figure 2.1 and summarized in the right column of Table 2.1.1. Production rate (tonnes/day) x(1). Production rate is one of the most important operating considerationsin pulp mills. The production rate can be changed by adjusting the chip transfer screw speed. However,the production rate varies with the variations in the raw wood chip quality such as wood species, thechip density, and the moisture content. The production rate can also affect the SE and the pulp quality.2. Primary/secondary motor load (MW), x(2), x(4). Among the operating variables, motor load is one ofthe most important measurements which is highly correlated with pulp qualities. One way to set andmaintain the motor load is to adjust the plate gap since motor load is directly affected by and sensitiveto the gap size.3. Primary/secondary consistency (%), x(3), x(5). At a given SE, consistencies in the refining zone forboth the primary and secondary refiners have a major influence on pulp properties. An improvedconsistency control can significantly reduce the fluctuations of the motor load and enhance the refinerperformance.Steam production is also an important operating variable since it is associated with high energy consumption.However, as is common practice in most pulp mills, this variable is rarely considered in controller design forMP processes.Specific energySpecific energy (MW/tonnes/day) (SE) is the energy consumed per ton of dry pulp and is a critical variablethat strongly affects pulp properties [32]. SE is defined as the ratio of the motor load with respect to theproduction rate. Therefore, the desired SE can be achieved by adjusting the motor load and production rateproperly. One of the traditional pulp quality control strategies is to manipulate SE based on the quality-energy relationship. For the i-th refiner and motor, SE is defined as follows:SE :=Motor loadiProduction rate, i = 1,2, (2.1)where the subscript index i = 1,2, represents the primary and secondary HC refiners, respectively. For atwo-stage HC MP process, the total specific energy (TSE) for both primary and secondary refiner is defined142.2. Key Process Variablesas,TSE :=Total motor loadProduction rate. (2.2)In our econ MPC design, TSE is embedded directly in the objective function and is used as an indicator ofeconomic performance.2.2.3 Pulp PropertiesThe final pulp quality can be characterized by many property variables such as freeness, fibre length, shivecontent, coarseness, and strength. To assess the quality of the pulp, we consider the following commonlyused pulp properties: Canadian Standard Freeness (CSF, ml), long fibre content (LFC, %) and shive content(SC, %). Such choice of the pulp properties is reasonable from the controller design point of view since thehandsheet strength and pulp drainage properties can be predicted using these variables. Moreover, these pulpproperties can be measured online using the available automated measuring devices. The detailed definitionand description of these pulp properties are as follows,1. CSF is a measure of the volume of water collected from a pulp suspension drained from one exit-nozzle in a special dewatering cell [60]. Freeness measures the degree of the refining applied on thepulp. Generally, the higher freeness implies that the pulp is easier to be drained.2. LFC, which is a variable describes the strength of the pulp, is another important pulp property that weconsider. The longer the individual fibre is, the stronger the pulp that one can get.3. SC, which is defined as the percentage of oven-dry pulp retained on a standard slotted fractionatingplate, is another indicator of the pulp strength and drainage properties [33].2.2.4 Process disturbancesWood chips are the main raw materials for the pulp production and thus variations in wood chips comprisethe main disturbance that affects the refining conditions and final pulp properties. In this thesis, variationsexisting in the raw chips (such as the chip bulk density and chip moisture content) will be considered asdisturbances.152.3. Two-stage HC MP Process Model2.2.5 Unmeasurable state variablesThe consistency in the primary and secondary refiners (x(3) and x(5)) cannot be measured reliably in real-timeand thus need to be inferred from the measured outputs (x(1), x(2), and x(4)) and inputs (u(1), . . . , u(5)). Theconsistency in the refiner is known to significantly impact pulp properties. A variety of advanced controlstrategies have been proposed in the literature under the assumption that the consistency can be measuredaccurately and that it can be used directly by the controller. However, such assumption becomes questionablegiven that practical sensors are usually not fast enough to measure the consistency in real-time. As a result,a reliable state estimation mechanism for the consistency is needed to reduce fluctuations in the motor loadand for stabilizing the MP process.2.3 Two-stage HC MP Process Model2.3.1 Nonlinear MP Process ModelMP processes are inherently MIMO processes with complex dynamics and interactions among process vari-ables. Modeling of MP process is challenging due to the complex mechanism inside of the pulp refiners.In this section, the two-stage HC MP process model is presented. The mathematical model is developedby using a combination of mechanistic and empirical methods, which will not only give some insights intomechanism, interactions, and nonlinearity of the MP refining process, but also characterize the feature of MPprocess dynamics. In this paper, the following process variables are used to develop a discrete-time nonlinearmodel for the MP process [43–46, 92–94].Production rateP = ka · kp · sc ·dc ·R, (2.3)where P (tonnes/day) is the production rate. ka and kp (m3/rev) are constant parameters which can beobtained from the industrial data and their values depend on the particular production lines. sc(%) is the chipsolid content. dc (kg/m3) is the chip bulk density. R (rpm) is the chip-transfer screw speed.Motor loadMi =kmi ·PDi(1− e(−10Gi))(ci− ei ·Gi), i = 1,2, (2.4)162.3. Two-stage HC MP Process Modelwhere Mi (MW ) is the motor load for the i-th refiner, i = 1,2. Di (l/min) is the dilution water flow rate. Gi(mm) is the gap distance. ci, ei, and kmi are the parameters of each refiner.ConsistencyCp =100PP+ ka ·Dp− kep ·Mp, (2.5)Cs =100PP/(0.01Cp)+ ka ·Ds− kes ·Ms, (2.6)where Cp and Cs are the consistency for the primary and secondary refiner, respectively. ka, kep and kes arethe refiner parameters.By introducing linear dynamics for the discretized differential state variables and superimposing it on thesteady-state relationships (2.3)–(2.6), a discrete-time nonlinear model for the MP process can be formulatedat sample time t with the state variables and manipulated input variables defined in Table 2.1. One can usethe time constant and time delay information of each subprocesses to form the dynamic matrix A as follows[43],A¯(z) =g1(z) 0 0 0 00 g2(z) 0 0 00 0 g3(z) 0 00 0 0 g4(z) 00 0 0 0 g5(z), (2.7)where A¯(z) is the dynamic transfer function matrix of the MP process. g1(z) is the transfer function betweenthe production rate and the chip-transfer screw speed. g2(z) is the transfer function between the primarymotor load and the primary refiner gap. g3(z) is the transfer function between the primary consistency andthe primary dilution flow rate. g4(z) is the transfer function between the secondary refiner motor load andthe secondary refiner plate gap. g5(z) is the transfer function between the secondary consistency and thesecondary dilution flow rate. The transfer functions gi(z), i = 1, . . . ,5, have the following forms,gi(z) =biz−diz−ai , i = 1, . . . ,5, (2.8)172.3. Two-stage HC MP Process Modelwhere ai, bi = 1−ai, i= 1, . . . ,5, are parameters for unity dynamic gains of each subprocess. di, i= 1, . . . ,5,are time delays of the subprocess. Note that the parameters ai, di, and bi will vary with the different refinersin each pulp mill. Then the dynamic matrix can be expressed as,A = diag{a1,a2,a3,a4,a5}. (2.9)By superimposing the nonlinear steady-state functions (2.3)–(2.6) to the linear dynamics of the two-stageHC MP process (2.7)–(2.9), the nonlinear two-stage HC MP process can be described as follows,x(1)t+1 = a1x(1)t +b1ka · kp · sc ·dc ·u(1)t−d1 , (2.10a)x(2)t+1 = a2x(2)t +b2km1 · x(1)tu(3)t(1− e−10u(2)t−d2 )(c1− e1 ·u(3)t ), (2.10b)x(3)t+1 = a3x(3)t +b3100x(1)tx(1)t + ka ·u(3)t−d3− kep · x(2)t, (2.10c)x(4)t+1 = a4x(4)t +b4km2 · x(1)tu(5)t(1− e−10u(4)t−d4 )(c2− e2 ·u(4)t−d4), (2.10d)x(5)t+1 = a5x(5)t +b5100x(1)tx(1)t /(0.01x(3)t )+ ka ·u(5)t−d5− kes · x(4)t, (2.10e)where the notation x(i)t , u(i)t , i = 1, . . . ,5, t ≥ 0, are the i-th state or manipulated variables (defined in Table2.1) at sampling time t, respectively.2.3.2 General Process ModelConsidering the variations in wood chips, such as the bulk density (dc) and the solid content (sc), in the restof this thesis, we use the following general nonlinear difference equations to describe the two-stage HC MPprocess,xt+1 = Axt +h(xt ,ut ,ζt) = f (xt ,ut ,ζt), (2.11a)yt = g(xt)+ηt , (2.11b)where A ∈ Rnx×nx , as defined in (2.9), is the dynamic matrix which can be identified for the MP processby using linear system identification methods. xt ∈ Rnx , ut ∈ Rnu , and yt ∈ Rny are the states, manipulated182.4. Pulp Quality Modelingvariables, and controlled outputs, respectively. For the two-stage HC MP process under study, xt and ut aredefined in Table 2.1. The state and input variables are required to satisfy the constraints xt ∈ X and ut ∈ U,where the sets X ⊆ Rnx and U ⊆ Rnu are compact and contain the equilibrium point (xss,uss). ζt ∈ Rnζand ηt ⊆ Rnη are the model uncertainty and measurement noise, respectively. h(·) : Rnx ×Rnu 7→ Rnx is anonlinear state function. f (·) : Rnx ×Rnu 7→ Rnx is a nonlinear function which represents the dynamics thatmap the current inputs and states to the states at the next time instant.Remark 2.3.1. Even though no work has been reported in the literature on the comparison in accuracybetween linear and nonlinear models of the two-stage HC MP processes. However, numerous strategies havebeen proposed for the controller design of MP process using linear models [29, 32] and nonlinear models[43, 45, 46], respectively.2.4 Pulp Quality ModelingIn this thesis, we use the nonlinear model developed in [46, 78] to predict the pulp properties of CSF, LFC,and SC. The following definitions are introduced before we establish the nonlinear pulp property models.Refining intensityThe refining intensity (RI) has been suggested as an important variable in all types of refining [32]. For agiven SE, different RI will produce the pulp with quite different quality.RIi =SE iNi, i = 1,2, (2.12)where SE i, i = 1,2, is the SE for the i-th refiner as defined in (2.1). Ni is the total number of impacts and isgiven by,Ni = nihiωi[(r1i+ r2i)/2]τi, i = 1,2, (2.13)where ni is the number of bars per unit length of arc of a refiner disc. hi = 1 for a single disc refiner, andhi = 2 for a double disc refiner. ωi(radians/s) is the refiner rotational speed. r1i, r2i(m) are the inlet andoutlet radius of plates’ refining zone. τi(s) is the residence time of the wood chips in the i-th refiner.192.5. SummarySpecific refining powerThe specific refining power (SRP) describes the energy-transfer rate. For the i-th refiner, SRPi is defined as,SRPi =SE iτi, i = 1,2. (2.14)The empirical relationships between pulp properties of CSF, LFC, and SC and the intermediate variablesSEi, RIi, and SRPi can be expressed as [46],CSFi = [CSFi0− kcs f1i (SEi−SEi0)] ·10−kcs f2i (RIi−RIi0), (2.15a)LFCi = LFCi0− kl f c1i (SRPi−SRPi0)− kl f c2i (SEi−SEi0), (2.15b)SCi = SCi0 ·10−[ksc1i (SEi−SEi0)+Ksc2i (SRPi−SRPi0)], i = 1,2, (2.15c)where CSFi0, LFCi0, and SCi0 are the initial values of pulp properties. SEi0, RIi0, and SRPi0 are the initialvalues of the SEi, RIi, and SRPi, respectively. kcs f1i , kcs f2i , kl f c1i , kl f c2i , ksc1i , and ksc2i are parameters of theprimary and secondary refiners and may vary with the different refiners in each pulp mills.2.5 SummaryIn this chapter, we provide a detailed description of the key process variables, including the manipulatedvariables, state and operating variables, and pulp properties. Based on these variables and the data collectedfrom the industry, a two-stage HC MP process model is provided. To simplify the notations for the controllerand estimator design in rest of this thesis, we provide a general form of the nonlinear process model.20Chapter 3Model Predictive Control3.1 IntroductionMPC, which is also known as the moving horizon or receding horizon control, is an online optimization-based control technique for constrained MIMO systems [17, 68, 79]. MPC has been extensively implementedin the industry as an effective approach to deal with large multi-variable industrial processes. The main ideaof MPC is determining control actions by repeatedly solving an online constrained optimization problem,which aims at minimizing a performance index over a finite prediction horizon. The first input of the optimalinput sequence is then injected into the plant and the problem is solved again at the next sampling time withupdated process measurements and a forward shifted horizon.The content of this chapter is outlined as follows. The traditional tr MPC is introduced in Section 3.2,in which the basic concepts and detailed steps of the tr MPC formulation for a linear state space model arepresented. Section 3.3 is devoted to the introduction of econ MPC, in which the existing research resultson econ MPC, m-econ MPC, and SMPC are provided. In Section 3.4, we demonstrate the detailed steps onhow to formulate a MPC for a nonlinear system. Some of the important concepts and definitions are alsoincluded. The stability analysis of NMPC is given in Section 3.5. The nonlinear solver IPOPT is explainedin Section 3.6, followed by the summary of this chapter in 3.7.3.2 Traditional Tracking MPCThe concept of MPC is depicted in Figure 3.1 and summarized in Table 3.1. To be specific, assuming thatthe current time is labeled as time instant t, at each sample time the current system measurements can beobtained either by online measurements or a state estimator. The future output trajectory over a predictionhorizon Nmpc, as shown in Figure 3.1, can be predicted based on the internal process model. The optimalinput sequence over a control horizon is calculated by minimizing an objective function subject to a set of213.2. Traditional Tracking MPCt 1t  2t   mpct N 1mpct N FuturePastPredicted trajectoryPredicted inputApplied inputActual trajectorySetpoint1t 2t Figure 3.1: Basic concept of MPCconstraints. The first move of the calculated future control input sequence is then applied to the process untilnext sample time. At the next sample time, the control sequence can be determined in a similar fashion withthe new process states and outputs. Performing the above procedures repeatedly as depicted in Table 3.1forms the logic of the receding horizon scheme. The repetitive nature of the MPC at each sampling timeleads to an implicit feedback control.We firstly consider a linear discrete time system at sampling time t described by a state space model as,xt+1 = Axt +But , t ≥ 0, (3.1a)yt = Cxt , (3.1b)subject to the control and state constraints,ut ∈ U and xt ∈ X, (3.2)where A, B, and C are the system matrices. xt ∈ Rnx and ut ∈ Rnu are the state and manipulated inputvariables, respectively. U ∈ Rnu and X ∈ Rnx are the constraint set for the manipulated and state variables,respectively.223.2. Traditional Tracking MPCTable 3.1: Basic steps of MPCAlgorithm 3.1Step 1: Given the current state measurements xt at time t.Step 2: Solve the constrained optimization problem and calculate a sequence of optimal future control inputs.Step 3: Apply the first control input of the optimal control sequence to the process.Step 4: At next sampling time, repeat Steps 1–3.One basic formulation of MPC based on state space model (3.1a)–(3.1b) is as follows,minv0|t ,...,vNmpc−1|tNmpc−1∑k=0∥∥zk|t − xss∥∥2Qx +∥∥vk|t −uss∥∥2Qu , (3.3a)s.t. z0|t = xt , (3.3b)zk+1|t = Azk|t +Bvk|t , k = 0, . . . ,Nmpc−1, (3.3c)vk|t ∈ U, k = 0, . . . ,Nmpc−1, (3.3d)zk|t ∈ X, k = 1, . . . ,Nmpc, (3.3e)where Nmpc is the prediction horizon in sampling times. zk|t , k = 0, . . . ,Nmpc and vk|t , k = 0, . . . ,Nmpc− 1,denote internal controller state and input variables predicted at time t for k steps into future. The solutionof this optimization problem depends on the current system state xt (3.3b). xss ∈ Rnx and uss ∈ Rnu are thesteady-state and optimal input which can be obtained by the static optimization or from the upper layer asdepicted in Figure 1.2. Qx ∈ Rnx×nx and Qu ∈ Rnu×nu are positive definite weighting matrices for stage andcontrol vectors, respectively. The decision variable is defined as ut = {v0|t ,v1|t , . . . ,vNmpc−1|t}, where theindex Nmpc is the control horizon specifically. However, we assume that the length of control horizon equalsto the prediction horizon and we use Nmpc as a notation for both the control and prediction horizon throughoutthis thesis. At each time instance t, only the first element of the optimal control sequence is applied to theplant, i.e.ut = v¯0|t . (3.4)The stage cost for the tr MPC is denoted as Ltr : Rnx×Rnu 7→ R, which is defined as follows,Ltr =∥∥zk|t − xss∥∥2Qx +∥∥vk|t −uss∥∥2Qu . (3.5)The wide application of MPC in industry is because of its strengths in a). handling MIMO systems, b).233.3. Economic MPCdealing with hard constraints directly in its framework, and c). aiming at the optimal performance by solvingonline optimization problems. The performance of MPC significantly relies on the accuracy of the processmodel. Thus, a sound nonlinear model can improve the performance of MPC controller with extra compu-tational cost which is induced by the model nonlinearity and the non-convexity of the optimization problem.However, the solution of the underlying nonlinear optimization problem is computational time-consuming.Hence, NMPC is only applied in relatively slow processes. Recently, with advanced computation hardwareand optimization techniques, NMPC can be extended to a large application area. Another disadvantage ofMPC is that it requires directly available full state information to solve the optimization problem. Otherwise,a state estimator is needed for the unmeasurable states.3.3 Economic MPC3.3.1 Economic MPC (Econ MPC)The goal of the optimization-based control is to achieve maximized net returns considering model uncer-tainties, measurement noises, and/or disturbances. In the traditional MPC, this objective is achieved in twolayers as illustrated in Figure 1.2. Specifically, the process economics are solved in the RTO layer to obtainthe economically optimal steady-state setpoints. In the advanced control layer, the economic objective isconverted into the control objective, based on which the controller is designed such that the steady-statesetpoints are tracked. Thus, the controller in the advanced control layer generates a solution that drives thesystem to the economically best steady states. However, the economic information is significantly lost wheneconomic objectives are converted into control objectives, since the controller in the advanced control layeris unaware of these possible high profit states in the global economic regions [6]. As a result, the hierarchicalseparation of the two-layer control structure in the traditional MPC design is no longer optimal or desirable.The econ MPC has been intensively studied as a standard method that merges the steady-state optimizationinto the dynamic MPC layer [37, 58, 75].Let t be current time stamp, the setpoint tracking value function denoted as V trt for tr MPC and the243.3. Economic MPCeconomic value function denoted as V ect for econ MPC are defined respectively as follows,V trt :=Nmpc−1∑k=0Ltr(xk|t − xss,uk|t −uss), (3.6)V ect :=Nmpc−1∑k=0Lec(xk|t ,uk|t), (3.7)where Ltr and Lec represent the tracking stage cost and economic stage cost, respectively. The mappingLtr : Rnx ×Rnu 7→ R is always nonnegative, and Ltr = 0 holds if and only if xt = xss and ut = uss. In thesequel, we use Ltr(xk|t ,uk|t) to denote the stage cost Ltr(xk|t − xss,uk|t − uss) for simplicity. The economicstage cost Lec : Rnx ×Rnu 7→ R is bounded and related directly to the desired economics (not necessarilydependent on the steady-state (xss,uss)). For brevity, the trajectory of xk|t , and uk|t , k = 0, . . . ,Nmpc− 1, isdenoted by {xk|t ,uk|t}Nmpc−1k=0 in the rest of this thesis.3.3.2 Multi-objective Economic MPC (M-econ MPC)In the past few years a number of econ MPC algorithms have been proposed to reduce the electrical en-ergy usage for MP processes [43–46, 93]. However, the econ MPC is particularly difficult to apply on thetwo-stage HC MP process due to the severe process nonlinearity and conflicting objectives in tracking per-formance and in minimizing energy costs. Clearly, exceptional tracking performance requires high energyinput and vice versa.Stability of the econ MPC has been a subject of active research in the last few years [7, 8, 31, 37, 42,47]. It is now known that the stability of general nonlinear econ MPC formulations cannot be establishedusing traditional Lyapunov settings [7, 8]. One strategy to circumvent this problem includes the addition ofregularization (convexification) terms guaranteeing that the regularized economic value function becomesa Lyapunov function [31, 47]. Alternatively, system-theoretic properties (e.g., dissipativity) can be used[8]. Recently, Zavala et al [103] interpreted the econ MPC as a multi-objective optimization problem thatseeks to simultaneously minimize the economic and the tracking performance. This interpretation allows theauthor to construct a stabilizing constraint that guarantees closed-loop stability for general economic MPCformulations.A constrained multi-objective optimization problem consists of optimizing multiple and often conflictingperformance criteria with a number of inequality and/or equality constraints. Multi-objective optimization253.3. Economic MPCapproaches are first proposed in the 1990s [52, 57, 86, 89], and further developed in [26, 27, 51, 65]. Themulti-objective formulation in the context of MPC is investigated in the past decade in [10, 22, 25, 64, 103,105]. In [10, 22] the control action is chosen among a set of Pareto optimal solutions. In [25], the optimalcontrol move minimizes the maximum of a number of performance indexes rather than seeking the Paretooptimal solutions in the standard multi-objective setting. In [64], the controller switches objective based onthe value of the state vector under stabilizing constraints. The authors in [105] propose a utopia-trackingstrategy to handle multiple conflicting objectives in MPC. The controller minimizes the distance of the costfunction to that of the steady-state utopia point. In [103], a flexible stabilizing constraint is incorporated tothe econ MPC formulation which preserves stability of the auxiliary MPC controller.3.3.3 Stochastic MPC (SMPC)Most industrial process model involves some level of uncertainties either from exogenous disturbances ordue to the unknown dynamics [18]. MPC aims at determining an optimal control by solving an open-loopfinite horizon constrained optimization problem with respect to a performance cost at each sampling instant[17, 79, 97]. The performance cost specifies desirable targets to be achieved by the controllers, e.g., a smalldeviation between predicted outputs and the pre-specified setpoints. Traditional (deterministic) MPC doesnot admit a systematic way to deal with model uncertainties and disturbances. Most existing MPC techniquesoften either completely neglect model uncertainties/disturbances, or simply formulate them into a min-maxproblem in which the optimal inputs are calculated by minimizing the cost function under the worst possibledisturbance and uncertainty.According to the types of models used, MPC can be broadly classified into deterministic MPC (DMPC)and SMPC. DMPC, for both linear and nonlinear processes, usually assumes that the process model isaccurate and that the future disturbances are constant. However, such assumptions often lead to poor closed-loop performance when the system is impacted by random disturbances or has large model uncertainties.In contrast, SMPC accounts for such stochastic properties and has become a parallel line of research [19,28, 87], with typical applications to building climate control, wind turbine control, power generation anddistribution, and network traffic control, etc. [19, 21, 74, 77, 87, 100]. Due to the presence of randomvariables, both cost functions and constraints may become stochastic. SMPC usually assigns a presumeddistribution function to random disturbances or measurement noises, and optimizes the expected value of acost function with the constraints satisfied for all scenarios [84, 100].263.4. Nonlinear MPC (NMPC)A major issue associated with SMPC is computational tractability [50, 87, 88]. Tractability issues arisebecause the control formulation incorporates statistical measures for the objective and constraints (whichare random variables). For instance, in the most basic SMPC formulation, one seeks to minimize the ex-pected value of the objective function (this involves a high-dimensional integral). As a result, one needsto approximate the expectation using quadrature techniques such as Monte Carlo sampling or polynomialchaos expansions [69]. These transcription approaches convert the infinite-dimensional SMPC problem intoa standard (finite-dimensional) optimization problem that can be handled using off-the-shelf solvers. Un-fortunately, the resulting optimization problems are often computationally expensive (e.g., they may requiremany scenarios to approximate statistical measures). These computational tractability issues are exacerbatedin more sophisticated SMPC formulations in which one might seek to optimize complex statistical measures(e.g., variance, conditional value at risk, quantiles, medians). Under such formulations, as the resulting op-timization problems might involve integer variables, complex nonlinearities, or even bilevel formulations.Similar tractability issues arise when dealing with constraints; specifically, constraints for SMPC are oftenenforced using statistical measures such as chance constraints, quantiles, risk measures, and almost-surelyconstraints (i.e., constraints are satisfied for all scenarios) [87].One of the commonly used sampling-based SMPC techniques is the scenario-based SMPC (SSMPC).It uses sampled scenarios from the distributions of random uncertainties and disturbances to approximatetheir true probability distributions [15, 16, 87]. With the expected cost function approximated by the sampleaverage from these scenarios, SSMPC obtains the optimal input that satisfies the constraints under all thescenarios of uncertainties and disturbances. One issue associated with SSMPC is the overly sensitivity toextreme scenarios with low probability. Moreover, SSMPC uses the expected cost to seek the best solution,which implicitly assumes that the average behavior is of primary interest, but other features, such as thevariance, are not concerned.3.4 Nonlinear MPC (NMPC)MPC can be categorized with the selection of model and/or objective function. NMPC refers to the internalmodel used in the problem is nonlinear. Based on the type of selected objective function, NMPC can be fur-ther classified as setpoint tracking or economic NMPC. Recent advances in process economic optimizationand process control have been focused more on NMPC, which extends the traditional linear MPC internal273.4. Nonlinear MPC (NMPC)model to nonlinear models and non-convex cost functions for trajectory tracking and dynamic optimization[2, 55]. The standard linear MPC are easier to optimize since it normally involves convex quadratic objec-tives and linear system dynamics. As a comparison, NMPC optimizes nonlinear and normally non-convexobjectives, which pose a lot of problems and challenges. For example, the stability properties of traditionallinear MPC have been well studied and established, whereas there is less literature and research on the stabil-ity theory of NMPC. Moreover, since economic NMPC optimizes process economics instead of tracking theoptimal setpoints, it may also lead to a non-steady operation, which might not be acceptable from the con-trol operator’s point of view. However, the possibility of obtaining a more significant economic advantagemotivates researchers and control engineers to investigate NMPC strategies for nonlinear processes.Consider a discrete time nonlinear system at time t described by the following difference equations,xt+1 = f (xt ,ut), (3.8)and it is subject to the following control and state constraints,ut ∈ U and xt ∈ X, (3.9)where f : Rnx×Rnu 7→ Rnx is the nominal nonlinear system function. xt ∈ Rnx and ut ∈ Rnu are the state andcontrol input variables, respectively. U ∈ Rnu is the control constraint set, and X ∈ Rnx is the state constraintset. In the simplest form of the constraints, U and X are given by,U := {ut ∈ Rnu |umin ≤ ut ≤ umax}, (3.10a)X := {xt ∈ Rnx |xmin ≤ xt ≤ xmax}, (3.10b)Given the system dynamics defined in (3.8), the constraints in (3.9), and the current states xt as initial283.4. Nonlinear MPC (NMPC)conditions, at each sampling time t the NMPC optimization problem can be formulated as follows 1,minv0|t ,...,vNmpc−1|tV NMPCt (zk|t ,vk|t), (3.11a)s.t. z0|t = xt , (3.11b)zk+1|t = f (zk|t ,vk|t), k = 0, . . . ,Nmpc−1, (3.11c)vk|t ∈ U, k = 0, . . . ,Nmpc−1, (3.11d)zk|t ∈ X, k = 1, . . . ,Nmpc, (3.11e)where V NMPCt : Rnx×Rnu 7→ R is the cost function over the horizon Nmpc,V NMPCt (zk|t ,vk|t) =Nmpc−1∑k=0L(zk|t ,vk|t), (3.12)with L(zk|t ,vk|t) : Rnx×Rnu 7→ R as the stage cost.Since MPC controller is working in a receding horizon fashion, the solution of NMPC (3.11a)–(3.11e) ateach sampling time t is denoted by u¯t = {v¯0|t , v¯1|t , . . . , v¯Nmpc−1|t}. The NMPC control law κnmpc(xt) : Rnx 7→Rnu can be defined by,κnmpc(xt) = v¯0|t . (3.13)In the nominal case, the control law drives the state of the plant towards xt+1 = zt+1 = f (xt ,κnmpc(xt)), wherezt+1 is the nominal model prediction.Definition 3.4.1. (Admissible set, admissible initial state set, admissible state set) In the MPC literature,let UN be the set of feasible control moves which can be obtained by solving (3.11a)– (3.11d) with initialstate x0. Then the admissible set WNmpc is defined as a joint set comprised of the initial states x0 and thefeasible input sequence ut = (v0|t ,v1|t , . . . ,vNmpc−1|t). To be specific,WNmpc := {(x0,ut)|∃ x1, . . . ,xNmpc : xt+1 = f (xt ,ut),xt ∈ X,ut ∈ UN}. (3.14)1In this thesis, we only consider the regulation problem, which is to bring the system state back to its fixed operating point froman initial point, for simplicity.293.5. Stability of NMPCThe corresponding set of admissible initial states ZNmpc is then defined as the projection ofWNmpc ,ZNmpc := {x0|∃ ut : (x0,ut) ∈WNmpc}. (3.15)The set of admissible states XN ⊆ X can be defined as,XN = {x|UN 6= 0}. (3.16)Definition 3.4.2. (Positively invariant set, output admissible set) Given the nonlinear system (3.8) withthe control law defined by u = κnmpc(x), and the initial condition f (0,0) = 0, the closed-loop system thencan be described as,xt+1 = f (xt ,κnmpc(xt)) = h¯(xt), (3.17)the set Xpi is positively invariant ifh¯(xt) ∈ Xpi, ∀xt ∈ Xpi.Moreover, if Xpi ∈ XN , then Xpi is called output admissible set, which is a attraction domain of the origin.3.5 Stability of NMPCClosed-loop stability is essential for a good control performance as well as the safety of the process. Intraditional the MPC formulation as shown in (3.3a)–(3.3e), the optimization uses a linear system model(3.1a)–(3.1b) and a quadratic objective function (3.3a). This leads to a convex quadratic optimization whichcan be quickly solved to obtain its unique global optimum. As is pointed out in [31] [103], in the trackingMPC (tr MPC), the stage cost, which is defined as the deviation from the best steady-state, is a positivedefinite function. The stability thus can be proved by treating the tracking stage cost as a Lyapunov function.However, the stability cannot be guaranteed when the stage cost is replaced by an arbitrary economic cost.This is because the economic cost may not be a positive definite function, and thus not qualified to be aLyapunov function.Remark 3.5.1. According to the Bellmann’s Principle of Optimality [9], when an infinite prediction andcontrol horizon is used in the optimization problem formulation, i.e. Nmpc → ∞, the predicted input andstate trajectories from the open-loop optimization problem (3.11a)–(3.11e) are the same as the closed-loop303.5. Stability of NMPCtrajectories of the nonlinear system. This indicates that system trajectory can be steered to the origin underthe NMPC control law calculated by solving (3.11a)–(3.11e) when Nmpc→ ∞, i.e., the origin of the closed-loop system under NMPC control law is asymptotically stable. In practical applications, the infinite horizonis normally approximated by a very large horizon to achieve the stability. However, for nonlinear system,the solution of large horizons in the nonlinear optimization problem is extremely difficult or impossible toobtain, which makes infinite horizon for NMPC only exists in a conceptual sense.Even though there is not a systematic way available to analyze the closed-loop stability based on theplant model, the objective function, and the horizon lengths, there are approaches that one can used sothat the stability of the closed-loop system can be guaranteed. Those approaches can be classified intofour configurations [68]. For more results on the convergence and stability of nonlinear econ MPC, see[6–8, 31, 42].1. Terminal equality constraint. In the terminal equality constraint configuration, equality constraintszNmpc|t = xss is added in the NMPC problem formulation (3.11a)–(3.11d). For simplicity, it is normallyassumed that xss = 0 in most existing literature. Specifically, it is called zero state terminal equalityconstraint in the case where xss = 0. The equality constraints require that the state vector needs toreach to the equilibrium point xss at the end the prediction horizon. This is the most widely suggestedNMPC scheme with guaranteed stability property. However, the equality constraint makes the non-linear optimization problem hard to solve in general. Thus, the implementation of terminal equalityconstraint configuration NMPC is limited [53].2. Terminal constraint set. A terminal set constraint zNmpc|t ∈ X f (no terminal cost) is added, whichforces the terminal state zNmpc|t to stay inside the neighborhood of the origin X f . The terminal regionX f ⊆X is compact and contains a neighborhood of the steady state xss. With this formulation, NMPCwill steer the system into a subset of the state-space that contains the steady-state xss. A dual modecontrol scheme is implemented in this configuration in which two different controllers are applied. Ifthe state is outside of the terminal region X f , a NMPC controller is applied. It is switched to a locallinear state feedback controller if the current state lies inside of the terminal region, i.e.ut =v¯0|t , if xt 6∈ X f ,Kxt , if xt ∈ X f .313.5. Stability of NMPC3. Terminal cost function. An additional terminal cost term is added to the objective function to achievethe nominal stability of the closed-loop system (3.17) [82]. The terminal cost function configurationis generally formulated as follows,V NMPCt = minv0|t ,...,vNmpc−1|tNmpc−1∑k=0L(zk|t ,vk|t)+F(zNmpc), (3.18a)s.t. z0|t = xt , (3.18b)zk+1|t = f (zk|t ,vk|t), k = 0, . . . ,Nmpc−1, (3.18c)vk|t ∈ U, k = 0, . . . ,Nmpc−1, (3.18d)zk|t ∈ X, k = 1, . . . ,Nmpc, (3.18e)where F(·) is a terminal cost or terminal penalty function. The cost function V NMPCt : Rnx ×Rnu 7→ Rcomprises the stage costs ∑Nmpc−1k=0 L(zk|t ,vk|t) and a terminal penalty function F : Rnx 7→ R.4. Terminal cost and constraint set. This NMPC configuration combines the aforementioned terminalcost and constraint set methods. This type of NMPC configuration attracts most attention in currentMPC research. In this configuration, a terminal set constraint zNmpc|t ∈X f is incorporated to the NMPCproblem formulation in (3.18a)–(3.18d). Many different choices of the penalty function F(·) and of theterminal set X f have been studied, such as infinite horizon, quasi-infinite horizon, zero-state terminalconstraint, to ensure the stability of the nominal closed-loop system (3.17) [20, 67, 70].Remark 3.5.2. The idea behind the aforementioned methods in order to enforce the closed-loop stability isto modify the NMPC problem setup so that the closed-loop stability can be guaranteed. Those modificationsare either adding suitable (equality or inequality) constraints or additional terms in the objective function tothe original NMPC problem. These addition constraints, which are usually termed as stability constraints,are of no actual physical meaning and added solely for the purpose of ensuring the stability of the closed-loopsystem.3.5.1 Nominal Stability of NMPCIn order to ensure the stability of the closed-loop system, in the standard MPC theory, the cost functionis always constructed to include penalties on the deviation of the predicted state from the desired steadystate [17, 68]. For the setpoint tracking NMPC, a quadratic stage cost Ltr =∥∥zk|t − xss∥∥2Qx +∥∥vk|t −uss∥∥2Qu is323.5. Stability of NMPCgenerally used in the cost function. Without loss of generality and to simplify the expression in this chapter,we require that the steady-sate setpoints (xss,uss) in (3.5) satisfy xss = 0 and uss = 0.The tracking NMPC with terminal cost and constraint set configuration can be formulated as,Vt = minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0Ltr(zk|t ,vk|t)+F(zNmpc), (3.19a)s.t. z0|t = xt , zNmpc|t ∈ X f , (3.19b)zk+1|t = f (zk|t ,vk|t), k = 0, . . . ,Nmpc−1, (3.19c)vk|t ∈ U, k = 0, . . . ,Nmpc−1, (3.19d)zk|t ∈ X, k = 1, . . . ,Nmpc−1. (3.19e)In order to describe the stability properties of NMPC, the following definitions and assumptions of thenonlinear system model f (·), stage cost Ltr(·), terminal cost F(·), constraint sets U and X, and terminal setX f are used throughout of this chapter.Definition 3.5.1. (K function,K∞ function) A continuous function α : [0,a)→ [0,∞), a ∈ R≥0, is a classK function if it is strictly increasing and α(0) = 0. It is a classK∞ function if α(a)→ ∞ as a→ ∞.Definition 3.5.2. (K L function) A continuous function β : [0,a)× [0,∞)→ [0,∞), is a classK L functionif for each fixed s ∈ R≥0, β (·,s) is a classK function, and for each r ∈ R≥0, β (r,s)→ 0 as s→ ∞.Definition 3.5.3. (Lyapunov function) [63] A function V (·) is a Lyapunov function for the system (3.17) ifthere exist a set Ξ, andK∞ functions αv1(·), αv2(·), αv3(·) such thatαv1(x) ≤ V (x) ≤ αv2(x), (3.20a)∆V (x) = V (h¯(x))−V (x) ≤ −αv3(x), ∀x ∈ Ξ (3.20b)Assumption 3.5.1. f (·) is continuous. The origin (0,0) is an equilibrium point of the system (3.8) andf (0,0) = 0.Assumption 3.5.2. Ltr(·) is Lipschitz continuous with Lipschitz constant Lv. Ltr(0,0) = 0, and αL(||x||) ≤Ltr(x,u)≤ βL(||x||) where αL and βL areK functions.Assumption 3.5.3. U is compact, and X is closed. The origin (0,0) is an interior point of X×U.333.5. Stability of NMPCAssumption 3.5.4. X f ⊆ XN is compact and contains the origin in its interior, i.e. 0 ∈ X f .Assumption 3.5.5. [43] Given a local continuous control law κ f : Rnx 7→ Rnu , the following assumptionshold,1. κ f is Lipschitz continuous in X f with the Lipschitz constant Lκ f ,2. κ f (x) ∈ U, ∀x ∈ X f ,3. f (x,κ f (x)) ∈ X f , ∀x ∈ X f ,4. F( f (x,κ f (x))−F(x)≤−Ltr(x,κ f (x)), ∀x ∈ X f .Assumption 3.5.6. At each time instance t, all the state variables xt are available for measurement.Assumption 3.5.7. No persistent disturbances act on the system (3.8).Assumption 3.5.8. The model used to predict the dynamic trajectory is accurate, i.e. there is no model-plantmismatch.Remark 3.5.3. Some of these aforementioned assumptions are only needed for some specific NMPC schemesin the following chapters. For example, Assumptions 3.5.6–3.5.7 are used in Chapters 3, 4, and 6. However,it is more practical to exclude some of these assumptions in Chapter 5, in which a state estimator is designedfor predicting the unmeasurable state variables in the pulp mill.Remark 3.5.4. The assumptions on the terminal constraint set X f , i.e. Assumption 3.5.4 and conditions 2and 3 in Assumption 3.5.5, indicate that X f is an output admissible set for the closed-loop system (3.17)under the local control law κ f . Condition 4 in Assumption 3.5.5 implies that the terminal cost F(·) decreasesalong the closed-loop trajectory f (x,κ f (x)).Lemma 3.5.1. [43] Given a local control law κ f , a terminal set X f , a terminal cost F, and a cost Ltr,satisfying the above assumptions 3.5.1– 3.5.5, the origin is an asymptotically stable equilibrium point forthe closed-loop system (3.17) with a region of attraction XN .Proof 3.5.1. The proof of Lemma 3.5.1 can be found in literature [63].Remark 3.5.5. Lemma 3.5.1 describes the nominal stability of the nonlinear closed-loop system (3.17). Thestability properties can be proved by using the classic Lyapunov function theory, in which the value functionVt is viewed as a Lyapunov function for the closed-loop system (3.17).343.5. Stability of NMPCRemark 3.5.6. It has been observed that the classical methodology for tracking MPC analysis cannot be ex-tended to the econ MPC when the nonlinearity is present. It is known that due to the potential non-convexityof the costs as well as the nonlinearity of the underlying dynamic system, the stability of general nonlinearecon MPC formulations cannot be established using traditional Lyapunov settings [7, 8, 31, 42]. One strat-egy to circumvent this problem includes the addition of regularization (convexification) terms guaranteeingthat the regularized economic value function becomes a Lyapunov function, as is done in [31, 47]. Anotheralternate approach is to rely on system-theoretic properties (e.g., dissipativity) to guarantee stability; thisapproach is discussed in [8]. Recently, Zavala et al [103] interpreted the econ MPC as a multi-objectiveoptimization problem that seeks to simultaneously minimize economic and tracking performance. This in-terpretation allows the author to construct a stabilizing constraint that guarantees closed-loop stability forgeneral econ MPC formulations.3.5.2 Robust Stability of NMPCLet the uncertain system be described byxt+1 = f (xt ,ut)+ `(xt ,ut ,ζt), xt ∈ X, ut ∈ U, ζt ∈W, (3.21)or equivalentlyxt+1 = f (xt ,ut ,ζt), xt ∈ X, ut ∈ U, ζt ∈W, (3.22)where f (xt ,ut) is the nominal system. ζt ∈Rnζ is the disturbance. W⊆Rnζ is the compact uncertain region.`(xt ,ut ,ζt) : Rnx ×Rnu ×Rnζ 7→ Rnx is the perturbation term which assumed to be Lipschitz continuous interms of all its arguments with Lipschitz constant L`.Given the NMPC control law κnmpc(xt) : Rnx 7→ Rnu as defined in (3.13), the closed-loop system can bewritten as,xt+1 = f (xt ,κnmpc(xt),ζt). (3.23)For the robust stability analysis in literature, the concept of Input-to-State-Stability (ISS) is most widelyused in literature. The following definitions, assumptions, and lemmas are used in this section to analyze therobustness of the NMPC controller.Assumption 3.5.9. The uncertain term `(·, ·, ·) is Lipschitz continuous w.r.t all its arguments with Lipschitz353.5. Stability of NMPCconstant L` ≥ 0 such that for all (x1,u1), (x2,u2) ∈ X, and ζ1, ζ2 ∈W the following equation holds,||`(x1,u1,ζ1)− `(x2,u2,ζ2)|| ≤ L`||(x1,u1,ζ1)− (x2,u2,ζ2)||. (3.24)Definition 3.5.4. (Admissible robust positive invariant (RPI) set) Given the system in (3.21) or (3.22) andthe closed-loop control law κnmpc, Xrpi ∈ XN is an admissible RPI set if the following hold,f (xt ,κnmpc(xt),ζt) ∈ Xrpi, ∀xt ∈ Xrpi, ∀ζt ∈W, (3.25)Definition 3.5.5. (Input-to-state stability, ISS) The closed-loop system (3.23) is ISS with ζ ∈W if thereexists aK L function β (·), and aK function γ(·) such that||xt || ≤ β (x0, t)+ γ(||ζt ||), ∀x0 ∈ Xrpi, ∀ζt ∈W. (3.26)Definition 3.5.6. (ISS-Lyapunov function) A function V (·) is called an ISS-Lyapunov function for system(3.23) if there exists a positive invariant set Ξ that contains the origin in its interior, K∞ functions αv1(·),αv2(·), αv3(·), and aK function δv1(·) such thatαv1(||x||) ≤ V (x) ≤ αv2(||x||), ∀x ∈ Ξ, (3.27a)V ( f (x,κnmpc(x),ζ ))−V (x,ζ ) ≤ −αv3(||x||)+δv1(||ζ ||), ∀x ∈ Ξ, ∀ζ ∈W. (3.27b)Especially, if the above conditions stand with δv1(·) = 0, then the origin is asymptotically stable for anydisturbance ζ ∈W.Lemma 3.5.2. [63] Given a admissible robust positive invariant set Xrpi that contains the origin, let V (·)be an ISS-Lyapunov function defined for the closed-loop system (3.23), then the system (3.23) is ISS in Xrpi.Proof 3.5.2. The proof of Lemma 3.5.2 can be found in literature [49, 63].Theorem 3.5.1. Under Assumptions 3.5.1–3.5.8, the closed-loop system (3.23) with closed-loop control lawκnmpc is ISS in an admissible RPI set Xrpi for any perturbation `(·, ·, ·) such that ||`(xt ,ut ,0)|| ≤ ρLvαl(||x||),where 0< ρ < 1 is an arbitrary real number, Lv is a Lipschitz constant of Vt defined in (3.19a), αl(·) is aKfunction.363.6. IPOPTThe detailed proof of Theorem 3.5.1 can be found in [63].Remark 3.5.7. The robustness analysis provided by the ISS framework is summarized in Theorem 3.5.1. Inthis section, the inherent robustness properties of nominal NMPC algorithm are discussed under the assump-tion that the feasibility of NMPC is not affected by the presence of uncertainties and disturbances.3.6 IPOPTIn general, the computational burden of NMPC is considerably higher than that of conventional MPC sinceNMPC may involve a non-convex cost function. Moreover, the nonlinear optimization problem has to becomputed online over a long prediction horizon at each sampling time for the sake of good closed-loop per-formances. Sometimes, the computational time required for solving the NMPC problem is even longer thanthe sampling time of the industrial process, especially when severe process nonlinearity and/or significantdisturbances are present. One remedy of this issue is to decompose the large scale systems into small sub-systems and apply distributed nonlinear MPC technique for a set of subsystems [5, 59, 85]. In recent years,some fast MPC algorithms such as Newton-type controller have been developed [99, 106]. To reduce thecomputational time, in this thesis, we formulate the nonlinear MP plant model in A Modeling Language forMathematical Programming (AMPL) and use IPOPT solver to solve the nonlinear optimization problem aspresented in [101, 104].Given a general form of the nonlinear optimization problem as shown in the following,minxz(x,θ), (3.28a)s.t. c(x,θ) = 0, (3.28b)x≥ 0, (3.28c)where x is the variable vector. z(·) and c(·) are the objective function and the quality constraints, respec-tively. θ is the parameter vector. In the context of NMPC formulation, θ ≡ x0 where x0 is the initial statevariable vector at each sampling time. IPOPT uses a primal-dual interior-point algorithm with a filter line-search strategy to solve this optimization problems. IPOPT computes a series of approximate solutions to a373.6. IPOPTsequence of barrier problems as follows,minxz(x,θ)−µn∑i=0ln(x(i)), (3.29a)s.t. c(x,θ) = 0, (3.29b)with a barrier parameter µ > 0. x(i) is the i-th component of the vector x. The logarithmic barrier terms,ln(x(i)), are added to the objective function in order to handle the inequality constraints in (3.28c). Asµ → 0, IPOPT calculates an approximate solution to the original problem by solving a sequence of barrierproblems. Given a fixed µ , IPOPT computes the following Karush-Kuhn-Tucker (KKT) conditions [43],∇xz(x,θ)+∇xc(x,θ)λ −ν = 0, (3.30a)c(x,θ) = 0, (3.30b)XVe−µe = 0, (3.30c)where λ is the Lagrange multiplier for the equality constraints. X = diag(x), V = diag(ν) are diagonalmatrices. e = [1, · · · ,1]T is a unit vector. As the barrier parameter decreasing, IPOPT applies a New-ton method to the KKT conditions (3.30a)–(3.30c). For a fixed barrier parameter µk and an iteration j,(x j(θ),λ j(θ),ν j(θ)), the Newton search direction (∆x j,∆λ j,∆ν j) can be obtained by solving the followingsparse KKT system,Wj A j −IATj 0 0Vj 0 X j∆x∆λ∆ν=−∆z(x j,θ)+A jλ j−ν jc(x j,θ)X jVje−µke (3.31)where Wj =∇xx(z(x j,θ)+c(x j,θ)Tλ j) is the Hessian of the Lagrangian functionz(x,θ)+c(x,θ)Tλ−νT .The matrix A j =∇xc(x j,θ) is the Jacobian of the constraint (3.30b). I and 0 denote identity and zero matriceswith appropriate dimensions. The IPOPT solver utilizes the basic formulations as presented above, readerscan also refer to [96] for more detailed descriptions.383.7. Summary3.7 SummaryIn this chapter, we present a basic formulation of MPC, which will act as a foundation knowledge for therest of this thesis. At the beginning, a brief introduction of the tradition tr MPC and econ MPC is presented.A mathematical representation of NMPC is given after the introduction. The nominal and robust stability ofNMPC, along with the basic definitions, assumptions, lemmas, are discussed. Moreover, the basic conceptand formulation of SMPC are introduced. Finally, the basic formulation of the computational solver IPOPT,which is used to solve the NLP of NMPC in this thesis work, is provided.39Chapter 4Econ MPC for a Two-stage HC MP Process4.1 IntroductionThe HC refining is one of the most energy intensive process in the MP process [32, 43]. Thus it is vital todevelop advanced control technique for the HC refining to reduce the energy consumptions in pulp mills.On the other hand, in order to ensure the stability of the closed-loop systems, in the standard linear MPCtheory, the deviation of the predicted state from the desired state is penalized. However, it has been observedthat the classic methods for the MPC analysis can not be extended to the econ MPC when nonlinearity ispresent [31]. It is known that due to the potential non-convexity of the costs as well as the nonlinearity ofthe underlying dynamic system, convergence can not always be achieved [7, 8, 42].In this chapter, we attempt to develop a nonlinear econ MPC algorithm for the two-stage HC MP process,as described in Chapter 1 and 2, to reduce the energy cost while ensuring the closed-loop stability and con-vergence. It is assumed that all the variables in the MP process are measurable. Specifically, in the proposedapproach, the econ MPC minimizes the TSE, which is related directly to desired economic considerations ofthe pulp mill and is not necessarily dependent on the steady state.The content of this chapter is constructed as follows. In Section 4.2, we demonstrate the econ MPCdesign for the two-stage HC MP process. To enforce convergence, two strategies with modified objectivefunctions in econ MPC schemes are proposed. Case studies are carried out in Section 4.3. In the firstsimulation, we compare the control performance of econ MPC with different penalty methods, while theenergy reduction is compared in the second case study. Section 4.4 summarizes this chapter.4.2 Econ MPC for the MP processThe models used in this chapter will be based on the ones derived in Chapter 2. The nonlinear two-stageHC MP process models are developed by using a combination of mechanistic and empirical methods. The404.2. Econ MPC for the MP processproduction rate, motor loads and consistencies for both primary and secondary refiners are treated as dis-cretized differential state variables. The chip-transfer screw speed, plate gap, and dilution water flow rates ofeach refiner are taken as manipulated variables. The model and disturbances are based on the data collectedin identification experiments on actual industrial processes, and it is shown that the developed model canrepresent the real process with high-fidelity. Based on this model, we will attempt to establish an econ MPCframework for the MP process to reduce the energy cost while preserving the closed-loop stability. Theeventual block diagram for the closed-loop MP process is illustrated in Figure 4.1.PredictionSetpoint calculationsControlcalculationsMPProcessMPModelPredictedoutputOptimalinputOptimalinputResidualsProcess outputModeloutput+-SetpointFigure 4.1: Block diagram for model predictive control of a typical MP processGiven the general nonlinear two-stage HC MP process model in (2.11a)–(2.11b), the econ MPC problemis formulated at sampling time t as follows,minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0Lec(zk|t ,vk|t), (4.1a)s.t. z0|t = xt , (4.1b)zk+1|t = f (zk|t ,vk|t ,ζk|t), k = 0, . . . ,Nmpc−1, (4.1c)yk|t = g(zk|t)+ηk|t , k = 0, . . . ,Nmpc, (4.1d)vk|t ∈ U, k = 0, . . . ,Nmpc−1, (4.1e)zk|t ∈ X, k = 1, . . . ,Nmpc, (4.1f)where the stage cost Lec(zk|t ,vk|t) := T SEk|t , k = 0, · · · ,Nmpc− 1. Nmpc is the prediction horizon, which isequal to the control horizon for simplicity. The current process state xt is assumed available at the current414.2. Econ MPC for the MP processtime t.Since the stability properties of the standard linear MPC theory cannot be extended to econ MPC es-pecially when the nonlinearity is present. Moreover, due to the non-convexity of the stage cost and thenonlinearity of the MP process dynamics, convergence is not always achievable. In this section, two differ-ent cases of econ MPC with modified objective functions are discussed.4.2.1 Case IFor the conventional MPC design, the steady-state xss and uss are usually obtained from the static optimizer,in which a linear model is assumed. In this section, a modified economic cost function is proposed as,V At := minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0Lec(zk|t ,vk|t)+∥∥zk|t − xss∥∥2QAx +∥∥∆vk|t∥∥2QA∆u , (4.2)where ∆vk|t = vk|t − vk−1|t , k = 1, · · · ,Nmpc − 1, is the increment of the manipulated variable. xss is thesteady-state which can be obtained by the static optimization or from the upper layer. QAx , QA∆u are positivesemidefinite weighting matrices.Remark 4.2.1. In order to guarantee the stability and the convergence of the closed-loop MP system, inscheme A, two convex terms are added to the original econ MPC cost function (4.1a). We should mentionthat, in [8], the authors only add the input increment term to guarantee the convergence. However, in aMP process, the final pulp property target of the production line may vary according to the customers’requirements. As a consequence, the steady states xss will change accordingly. Thus, the first convex termwhich penalizes the distance of the predict state xt from its steady state xss is also necessary in the objectivefunction.4.2.2 Case IIIn order to stabilize the system, another commonly used method is to add penalization on the distance ofmanipulated and state variables from their steady-state targets as a convex term in the objective function(4.1a) [8]. In this method, the modified econ MPC objective function is as follows,V Bt := minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0Lec(zk|t ,vk|t)+∥∥zk|t − xss∥∥2QBx +∥∥vk|t −uss∥∥2QBu , (4.3)424.3. Simulation Resultswhere xss and uss can be obtained by static optimization or from the upper layer. QBx , QBu are the weightingmatrices. Other variables and parameters are defined similarly as the ones in Scheme A.Remark 4.2.2. Note that even though one of the most important features associated with econ MPC techniqueis that it can directly optimize the economic cost of operating the plant without requiring any information ofthe steady state (xss,uss). This might render the precomputed steady state unnecessary. However, the workreported on econ MPC mostly includes the steady state information into the analysis. For example, the econMPC with a terminal constraint requires the state to be equal to the steady state at the end of the horizon [8];The strong duality or dissipativity assumption requires that the system have a priori knowledge of the steadystate in order to carry out the stability analysis [31].4.3 Simulation ResultsIn this section, we present two simulation examples to demonstrate the applicability of the proposed ap-proaches in MPC design for the nonlinear MP process. In Simulation I, we compare the control performanceby adding two different penalties in the economic objective functions as proposed in Scheme A and B. Inthe second simulation, the energy savings using the econ MPC are tested through a comparison with theconventional setpoint tr MPC.4.3.1 Simulation I: Comparison of Econ MPC with Different Penalty MethodsFor the two-stage HC MP process as defined in (2.11a)–(2.11b), the dynamic optimization problem with thepenalty terms in scheme A is formulated as follows,V At = minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0αLec(zk|t ,vk|t)+β{∥∥zk|t − xss∥∥2QAx +∥∥∆vk|t∥∥2QA∆u} (4.4a)= minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0α(z(2)k|t + z(4)k|t )/z(1)k|t +β{∥∥zk|t − xss∥∥2QAx +∥∥∆vk|t∥∥2QA∆u}, (4.4b)s.t. z0|t = xt , (4.4c)zk+1|t = f (zk|t ,vk|t ,ζk|t), k = 0, . . . ,Nmpc−1, (4.4d)xmin ≤ zk|t ≤ xmax, ymin ≤ yk|t ≤ ymax, k = 0, . . . ,Nmpc, (4.4e)umin ≤ vk|t ≤ umax, k = 0, . . . ,Nmpc−1, (4.4f)∆umin ≤ ∆vk|t ≤ ∆umax, k = 1, . . . ,Nmpc−1, (4.4g)434.3. Simulation Resultswhere zk|t , vk|t , and ∆vk|t , are the predicted state, manipulated, and the increment of the manipulated variablesin the two-stage HC MP process at time t for k steps in future. xss is the steady-state which can be obtainedfrom static optimization. f (·) is the nonlinear dynamic function of the process defined in (2.11a). Nmpc = 10is the prediction horizon which is assumed to be equal to the control horizon. xmin and xmax, umin and umax,ymin and ymax, ∆umin and ∆umax are the lower and upper bounds of the states, the manipulated variables, themeasured outputs, and the increments of the input, respectively. α and β are the weighting constants of theSE relative to the penalty terms. In this simulation, we specify that α = 1000, β = 1. It should be mentionedhere that, if the convergence speed is more important in the industrial applications or one wants to force thestates to reach the setpoints, more penalty should be added to β . The weighting matrices QAx and QA∆u arechosen to be QAx = diag([0.01,10,0.1,10,0.1]T ), QA∆u = diag([0.1,100,0.01,100,0.01]T ), respectively.0 10 20 30 40 50 60 70 80 90050100time (s)Secondary CSFml Scheme AScheme B0 10 20 30 40 50 60 70 80 9050556065time (s)Secondary LFC%0 10 20 30 40 50 60 70 80 900.40.60.81time (s)Secondary SC%Figure 4.2: Pulp quality after two-stage HC refining by using econ MPC in simulation 4.3.1444.3. Simulation ResultsAs a comparison, the modified econ MPC optimization in scheme B is as follows,V Bt = minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0αLec(zk|t ,vk|t)+β{∥∥zk|t − xss∥∥2QBx +∥∥vk|t −uss∥∥2QBu } (4.5a)= minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0α(z(2)k|t + z(4)k|t )/z(1)k|t +β{∥∥zk|t − xss∥∥2QBx +∥∥vk|t −uss∥∥2QBu }, (4.5b)s.t. z0|t = xt , (4.5c)zk+1|t = f (zk|t ,vk|t ,ζk|t), k = 0, . . . ,Nmpc−1, (4.5d)xmin ≤ zk|t ≤ xmax, ymin ≤ yk|t ≤ ymax, k = 0, . . . ,Nmpc, (4.5e)umin ≤ vk|t ≤ umax, k = 0, . . . ,Nmpc−1, (4.5f)where uss is the steady-state input which can also be obtained from the static optimization as xss in schemeA. QBx and QBu are the corresponding weights in scheme B. To have a fair comparison between scheme A andB, we specify QBx = QAx and QBu = QA∆u. The optimization constraints and the other variables and parametersin Scheme B are chosen to be the same as those of scheme A.In the closed-loop simulation, in order to demonstrate the economic benefits of the proposed econ MPCwith different penalty methods, we also considered the disturbances in state variables. The disturbances inthe refining process normally arise from the wood chip raw material and the refining process itself. Thedisturbances will affect the process operating conditions, thereby changing the final pulp quality. Figure4.2 demonstrates the pulp properties (CSF, LFC, and SC) after the two-stage HC MP process and the statetracking performance using econ MPC. As illustrated in Figure 4.2, the econ MPC with both the penaltymethods in Scheme A and Scheme B are able to ensure the pulp qualities to be within their quality boundswhich are 0∼ 400ml, 0∼ 100%, and 0∼ 4% for CSF, LFC, and SC, respectively. Besides, the two schemesdemonstrate a similar tracking performance of the state variables. In Figure 4.3 it can been seen that abetter tracking performance can be achieved by using econ MPC with the penalty in Scheme B. However,in Scheme A, instead of emphasizing the tracking effect of the manipulated variable, we only penalize theincrement of the input, which as a consequence, will give the nonlinear MPC controller leeway to find itsoptimal input that minimizes the cost function. The SE plot in Figure 4.5 further supports our analysis, inwhich extra 3.54% of the SE reduction is achieved by using Scheme A.454.4. Summary4.3.2 Simulation II: Energy Reduction by Using Econ MPC Compared with StandardMPCIn the standard MPC, the optimization problem, which is subject to constraints in (4.5c)–(4.5f), is defined inthe following form,V trt (x) := minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0||zk|t − xss||2Qtrx + ||vk|t −uss||2Qtru , (4.6)where Qtrx and Qtru are weighting matrices. In this simulation, a potential 18% of the SE reduction by usingecon MPC in Scheme A, and about 14% of the energy reduction can be achieved by using Scheme B. A moredetailed work about the energy reduction by using nonlinear MPC in comparison with the standard MPC canbe found in [46].4.4 SummaryIn this chapter, we investigate the econ MPC technique for a two-stage HC MP process. Two differentschemes are studied to guarantee the convergence and stability of the closed-loop nonlinear MP process. InScheme A, two convex terms, the deviation of the states from their targets and the increment of manipulatedvariables, are added in the objective function. In Scheme B, the deviation of both the state and manipulatedvariables are included in the cost function.In the simulation results, it can be also observed that some of the setpoints, for instance, the secondarymotor load (in Figure 4.3) and the screw speed (in Figure 4.4), are not reached. The controllers settle at asteady-state that is close to the desired one. The author is now looking into a better balance between trackingand economic without comprising the asymptotic stability. In the recent work [103], the author incorporateda stabilizing constraint to the econ MPC that preserves the stability of the auxiliary MPC controller. Based onthis new result, our next work would be applying the multi-objective optimization methods to the MP processthat can not only reduce the electrical energy consumption but also guarantee the asymptotic stability.464.4. Summary0 10 20 30 40 50 60 70 80 90100200300time (s)Production ratetonnes/day Scheme AScheme BSetpoints0 10 20 30 40 50 60 70 80 9001020time (s)Primary motor loadMW0 10 20 30 40 50 60 70 80 90204060time (s)Primary cosistency%0 10 20 30 40 50 60 70 80 90246time (s)Secondary motor loadMW0 10 20 30 40 50 60 70 80 90304050time (s)Secondary cosistency%Figure 4.3: The state variables of the MP process by using econ MPC in Simulation I474.4. Summary0 10 20 30 40 50 60 70 80 90102030time (s)Screw SpeedRPM Scheme AScheme BSetpoints0 10 20 30 40 50 60 70 80 90024time (s)Primary Gapmm0 10 20 30 40 50 60 70 80 90100150200time (s)Primary flow ratesL/min0 10 20 30 40 50 60 70 80 90024time (s)Secondary gapmm0 10 20 30 40 50 60 70 80 900100200time (s)Secondary Flow RatesL/minFigure 4.4: The manipulated variables of the MP process by using econ MPC in Simulation 4.3.1484.4. Summary0 10 20 30 40 50 60 70 80 9012001400160018002000220024002600time (s)Specific Energy, kWh/t Scheme A Scheme B Standard MPCFigure 4.5: Comparison of the energy reduction in Simulation 4.3.1 & 4.3.249Chapter 5Multi-objective Economic MPC for the MPProcess5.1 IntroductionThe ultimate goal of an econ MPC is to minimize the process energy costs while ensuring that the productmeets the minimum quality requirements. In many cases, direct minimization of the energy costs results inan unstable closed-loop MP system [44–46, 93]. There are several ways to ensure the stability of generalecon MPC, such as the traditional Lyapunov theory [31, 47] or the system-theoretic properties [8].The multi-objective optimization problem consists of optimizing multiple criteria which are usually con-flicting in terms of performance, and with a number of inequality and/or equality constraints. The m-econMPC interprets the econ MPC as a multi-objective optimization problem that simultaneously minimizes theeconomic and tracking performance. In this method, a flexible stabilizing constraint is incorporated to theecon MPC formulation in which the stability is preserved by the auxiliary MPC controller.Inspired by the m-econ MPC proposed in [103], in this chapter, we address the stability problem of theecon MPC problem for a nonlinear two-stage HC MP process by developing the m-econ MPC algorithm.The remainder of this chapter is outlined as follows: in Section 5.2, the m-econ MPC problem set-up and thestability analysis of m-econ MPC are provided. The m-econ MPC design for a two-stage HC MP processis given in 5.3. Simulation studies examining the performance of the proposed m-econ MPC technique areoffered in Section 5.4, followed by the summary of this chapter in Section 5.5.505.2. Multi-objective Control Problem Setup5.2 Multi-objective Control Problem Setup5.2.1 Basic NotationBefore proceed to the main content of this section, we revisit some of the notation that will be constantlyused in this chapter. Following the notation defined in Chapter 2, and assume that the optimal trajectory{x˜k|t , u˜k|t}Nmpc−1k=0 at time instant t is the solution of the following tr MPC problem,minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0Ltr(zk|t ,vk|t), (5.1a)s.t. z0|t = xt , zNmpc|t = xss, (5.1b)zk+1|t = f (zk|t ,vk|t ,ζk|t), k = 0, . . . ,Nmpc−1, (5.1c)vk|t ∈ U, k = 0, . . . ,Nmpc−1, (5.1d)zk|t ∈ X, k = 1, . . . ,Nmpc−1. (5.1e)Remark 5.2.1. As pointed out in [103] and [31], in the traditional tr MPC, the stage cost Ltr is a positivedefinite function. The stability thus can be proved by treating the tracking stage cost as a Lyapunov function.However, the stability cannot be guaranteed when the stage cost is replaced by an arbitrary economic costLec. This is because the economic cost may not be a positive definite function, and thus it is not qualifiedto be a Lyapunov function. For more results on the convergence and stability of the econ MPC, see [6–8, 31, 42]. Under the strong duality assumption, the authors in [31] propose a class of MPC schemes usingan economic cost objective that admits a Lyapunov function to establish the asymptotic stability propertiesof the closed-loop system. In [8], a terminal state constraint is used to force the predicted states to convergeto their steady states at the end of the horizon. This result is further relaxed in [42] to an asymptotic time-averaged econ MPC without terminal constraints, in which an approximate optimal performance is obtainedbased on certain controllability assumptions and on the turnpike property. A terminal region constraint isproposed in [7], and the stability is guaranteed by adding a penalty to the terminal state in the cost function.5.2.2 Stability of M-econ MPCAssume that {x˜k|t , u˜k|t}Nmpc−1k=0 is the optimal trajectory for the standard tr MPC problem (5.1a)–(5.1e). Asexplained above, the stability of the closed-loop system can be achieved by treating the tracking stage cost515.2. Multi-objective Control Problem SetupLtr as a Lyapunov function. Moreover, it has been proved that the stability can be guaranteed for any MPCcontroller generating a feasible trajectory {xk|t+1,uk|t+1}Nmpc−1k=0 as long as such trajectory satisfies:V trt+1 ≤ V˜ trt+1+σ(V trt −V˜ trt+1), (5.2)where σ ∈ [0,1) is a scalar. V˜ trt+1 := ∑Nmpc−1k=0 Ltr(x˜k|t+1, u˜k|t+1) is the value function of the optimal trajectory{x˜k|t+1, u˜k|t+1}Nmpc−1k=0 for the tr MPC. The condition (5.2) is also referred as the stabilizing constraint. Thefunction V trt is defined as,V trt :=Nmpc−1∑k=0Ltr(xk|t ,uk|t). (5.3)Since any feasible trajectory satisfying (5.2) is guaranteed to lead to stability, we can design an econMPC controller that enforces (5.2) directly. The formulation of this controller is given by:minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0Lec(zk|t ,vk|t), (5.4a)s.t. z0|t = xt , zNmpc|t = xss, (5.4b)zk+1|t = f (zk|t ,vk|t ,ζk|t), k = 0, . . . ,Nmpc−1, (5.4c)vk|t ∈ U, k = 0, . . . ,Nmpc−1, (5.4d)zk|t ∈ X, k = 1, . . . ,Nmpc−1, (5.4e)Nmpc−1∑k=0Ltr(zk|t+1,vk|t+1)≤ εt+1(σ), k = 0, . . . ,Nmpc−1, (5.4f)withεt+1(σ) := V˜ trt+1+σ(Vtrt −V˜ trt+1), (5.5)where σ ∈ [0,1).Notice that the constraint in (5.4f) is an equivalent expression of the stabilizing constraint in (5.2). Thespecific algorithm we propose to solve the above m-econ MPC problem is illustrated in Table 5.1. Recallthat in the m-econ MPC technique, the closed-loop stability is implied by the stabilizing constraint in (5.5).The proof of this statement is outlined as follows.Assume that the trajectory {x˜k|t+1, u˜k|t+1}Nmpc−1k=0 is an optimal trajectory for the standard tr MPC problem525.3. M-econ MPC Design for a Two-stage HC MP ProcessTable 5.1: The implementation of m-econ MPCAlgorithm 5.1Input: x0 ∈ X, σ ∈ [0,1), set t← 0 and ε0(σ)←+∞Output: for t = 0, . . . , simulation ends do1: Solve the m-econ MPC optimization in (5.4a)–(5.4f) forthe state xt and εt(σ), evaluate V trt , and set ut ← v¯0|t2: Implement ut to the plant and obtain the state variables xt+13: Solve tr MPC in (5.4f) for the state xt+1, and evaluate V˜ trt+14: Set εt+1(σ)← V˜ trt+1+σ(V trt −V˜ trt+1)5: end forin (5.1a)–(5.1e) at time t+1. Then the following inequality holds [68],V˜ trt+1−V trt ≤−Ltr(xt ,ut). (5.6)Adding the term −V trt to both sides of (5.2), we have,V trt+1−V trt ≤ (1−σ)(V˜ trt+1−V trt ). (5.7)Combining (5.6) and (5.7), the following inequality holds for any feasible solution,V trt+1−V trt ≤−(1−σ)Ltr(xt ,ut). (5.8)Since the function (1− σ)Ltr(xt ,ut) is nonnegative for σ ∈ [0,1), we can conclude that the trajectory{xk|t+1,uk|t+1}Nmpc−1k=0 obtained from the m-econ MPC algorithm is stable. It has been proved in [103] thatthe asymptotically stability with the region of attraction ZNmpc can be guaranteed under the the control lawobtained by solving the m-econ MPC problems for any σ ∈ [0,1).5.3 M-econ MPC Design for a Two-stage HC MP ProcessThe MP process is a complex MIMO nonlinear process with strong interactions among the variables. Basedon the MP model developed in Chapter 2 and the m-econ MPC approach proposed in previous sections,we are now in a position to apply the m-econ MPC technique to the MP process. For the MP process, the535.3. M-econ MPC Design for a Two-stage HC MP Processtracking and economic objective functions are defined to be,V trt =Nmpc−1∑k=0||xk|t − xss||2Qx + ||uk|t −uss||2Qu , (5.9)V ect =Nmpc−1∑k=0T SEk|t , (5.10)where Nmpc is the prediction horizon, Qx and Qu are positive-definite weighting matrices for the state andinput variables, respectively. T SEk|t is the total energy as defined in (2.2) and given by ([0,1,0,0,0] xk +[0,0,0,1,0] xk) / [1,0,0,0,0] xk for the specific MP process under study. In this chapter, all the state vari-ables in the MP process are assumed to be available at each sampling time t. The m-econ MPC optimizationfor the MP process can be formulated as follows,minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0T SEk|t , (5.11a)s.t. z0|t = xt , zNmpc|t = xss, (5.11b)zk+1|t = f (zk|t ,vk|t ,ζk|t), k = 0, . . . ,Nmpc−1, (5.11c)xmin ≤ zk|t ≤ xmax, ymin ≤ yk|t ≤ ymax, k = 0, . . . ,Nmpc, (5.11d)umin ≤ vk|t ≤ umax, k = 0, . . . ,Nmpc−1, (5.11e)Nmpc−1∑k=0Ltr(zk|t+1,vk|t+1)≤ εt+1(σ), k = 0, . . . ,Nmpc−1, (5.11f)withεt+1(σ) := V˜ trt+1+σ(Vtrt −V˜ trt+1), (5.12)where σ ∈ [0,1). εt+1(σ) is defined in (5.5). xmin and xmax, umin and umax, are the lower bounds and upperbounds of states and manipulated variables, respectively.Remark 5.3.1. In Chapter 3, we proposed an econ MPC algorithm for the MP process, in which two differentecon MPC schemes were investigated: one with penalty on the increment of the input and one with penaltyon the offset of the input from its steady-state [93]. It has been observed that in order to reduce the energyconsumption in terms of the SE, larger penalty has to be added to the economic term in the objective function.However, this may lead to a significant deviation of the state variables from the steady-state target. Similarissues have also been reported in [44–46]. This motivates our proposition of using the m-econ MPC to solve545.4. Simulation Resultssuch problems and achieve an acceptable compromise between the tracking performance and the economics.Remark 5.3.2. Note that by imposing the stabilizing constraint (5.11f) to the m-econ MPC, we can merge thecapability of tr MPC in assuring closed-loop stability and the merits of economic MPC together to achieveboth tracking and economic performances. Although we may sacrifice certain economic profit comparedwith the pure economic MPC, what we gain in guaranteeing the stability is much more meaningful andcrucial for most practical MP processes. Moreover, this m-econ MPC method provides a degree-of-freedom(the tuning parameter σ ) to tune the trade-off between economics and tracking performance.5.4 Simulation ResultsThe objective of this simulation is to show the effectiveness and the economical benefits of using the m-econMPC algorithm for the two-stage HC MP process. To be specific, we will show that the proposed m-econMPC algorithm can not only reduce the energy consumption in terms of SE, but also guarantee the closed-loop stability. This algorithm also allows the user to tune the controller in such a way that a desired trade-offbetween the economic and the tracking performances is achieved based on the practical demands.Table 5.2: The variations of chip bulk density dc and chip solid content sc of the raw material from theirnominal valuesTime (s) 0-50s 50-110s 110-160sChip bulk density (dc) 80% 115% 90%Chip solid content (sc) 90% 100% 110%In the closed-loop simulation, the variations in the raw materials such as the chip bulk density dcand the chip solid content sc are considered as the disturbances (see Table 5.2). The prediction hori-zon and the control horizon are selected to be equal and set to be Nmpc = 30. The sampling time is 2s,and the simulation length is 160. The weighting matrices Qx = diag([0.01,10,0.1,10,0.1]T ) and Qu =diag([0.1,100,0.01,100,0.01]T ). As mentioned in Section 4.4, the scalar σ has to be in the range [0,1) forclosed-loop stability. However, to demonstrate the effect of the parameter σ on the tracking performanceand the economics, here we allow σ = 1 and will examine the following four different values of σ : σ = 0,σ = 0.5, σ = 0.75, and σ = 1. Note that for σ = 0, the m-econ MPC will be reduced to the standard tr MPC.When σ = 1, it will be equivalent to the econ MPC without regulations. σ = 0.5 and σ = 0.75 are the twocases where we have the standard m-econ MPC.The simulation results are shown in Figure 5.1–Figure 5.4. From Figure 5.1, we can see that for these four555.4. Simulation Results0 20 40 60 80 100 120 140 1600200400Secondary CSFml σ =0 σ =0.5 σ =0.75 σ =10 20 40 60 80 100 120 140 16050607080Secondary LFC%0 20 40 60 80 100 120 140 160012Time in SamplesSecondary SC%Figure 5.1: Pulp qualities after two-stage HC refiningsituations, all of the pulp qualities such as the CSF, LFC, and SC, remain within their respective acceptableranges: 50−400ml, 50−80%, and 0−2%. However, by using the econ MPC (σ = 1), the pulp qualities aremore likely to hit the operating limits compared with the other two MPC schemes. This is not desirable fromthe perspective of mechanical pulping mills. Figure 5.2 and Figure 5.3 illustrate the tracking performanceof the state and manipulated variables. It can be seen that for σ = 0, σ = 0.5, and σ = 0.75, the state andmanipulated variables converge to the steady states but with different convergence speeds. Specifically, asσ decreases, the tracking speed of the m-econ MPC improves, which is consistent with our analysis sincesmaller σ values imply more emphasis on the tracking performance. For the extreme case where σ = 1,the convergence and stability cannot be guaranteed since the target in this case will be merely achieving theoptimal economic performance regardless of the tracking performance or even the stability.The comparison of the SE consumption between these four situations is illustrated in Figure 5.4. FromFigure 5.4, we can see that the m-econ MPC with σ = 0.5 and σ = 0.75 can save about 10% and 27% of theSE, respectively, compared with the tr MPC when σ = 0. Moreover, the econ MPC scheme where σ = 1shows the most amount of energy saved at about 73.12% but the closed-loop MP process is unstable.565.5. Summary5.5 SummaryIn this chapter, we present an m-econ MPC technique for a two-stage HC MP process. The stability ofthe closed-loop MP system is ensured by exploiting a stabilizing constraint and the inherent stability of thestandard tr MPC controller. The scalar σ associated with the m-econ MPC enables the trade-off betweenthe economics and the convergence rate of the tracking performance. From the simulations, we show thatthe m-econ MPC can be viewed as a more general scheme that includes the tr MPC and the econ MPC asspecial cases. In addition, the simulation results demonstrate the potential reduction in energy consumptionby using m-econ MPC.575.5. Summary0 20 40 60 80 100 120 140 160200300400Production ratetonnes/day Setpoint σ =0 σ =0.5 σ =0.75 σ =10 20 40 60 80 100 120 140 1600510Primary motor loadMW0 20 40 60 80 100 120 140 160204060Primary cosistency%0 20 40 60 80 100 120 140 1600510Secondary motor loadMW0 20 40 60 80 100 120 140 16020304050Secondary cosistency%Time in SamplesFigure 5.2: The state variables of the MP process585.5. Summary0 20 40 60 80 100 120 140 16010203040Screw SpeedRPM Setpoint σ =0 σ =0.5 σ =0.75 σ =10 20 40 60 80 100 120 140 160024Primary Gapmm0 20 40 60 80 100 120 140 160100200300400Primary flow ratesL/min0 20 40 60 80 100 120 140 160024Secondary gapmm0 20 40 60 80 100 120 140 16050100150200Secondary Flow RatesL/minTime in SamplesFigure 5.3: The manipulated variables of the MP process595.5. Summary0 20 40 60 80 100 120 140 1600200400600800100012001400Time in SamplesSpecific Energy, kWh/t σ =0σ =0.5σ =0.75σ =1Figure 5.4: Comparison of the energy reduction60Chapter 6MHE and the Integration of MHE andM-econ MPC6.1 IntroductionIn Chapter 5, we proposed an m-econ MPC for a two-stage HC MP process. We show that m-econ MPC cansimultaneously minimize SE cost while enforcing setpoint tracking (which is critical to maintain final pulpproduct quality). Recently, the inherent robustness properties for an economic NMPC controller is analyzedin [41], which provided a high flexibility to optimize economic performance and remains robust in the faceof disturbances. Despite these advances, extensive obstacles still prevent the deployment of NMPC in MPprocesses [44]. Among these obstacles, state estimation is of primary importance because many processvariables cannot be measured in MP mills.The majority of existing work on the advanced control for MP processes is based on the assumption thatstate variables are measurable and known in real-time. However, measurements on important variables suchas pulp consistency in HC refining are rarely available due to the lack of measurement sensors, particularlythose fast and reliable online ones. For instance, one of the most widely used sensors in pulping industryis the PQM which measures the shives, fibre size distribution, and freeness in the pulp. This sensor takesmeasurements every 50-60 minutes and thus, which limits the use of MPC. Thus, the state estimator designis of vital importance for the pulping industry.State estimation for nonlinear MP systems is particularly challenging, especially when there are con-straints on state variables [71, 72, 81]. To address such issues, MHE has been proposed as a practical ap-proach that can directly embed nonlinear dynamics and constraints [1, 81]. In MHE, the states are estimatedin real-time by solving a short horizon optimization problem that minimizes the difference between mea-surements and predicted outputs. A fixed size moving window slides forward in time and prior information616.2. State Estimator Designon estimated states is updated via the so-called arrival cost.In this chapter, we develop an MHE for the two-stage HC MP process and an econ MPC frameworkthat combines m-econ MPC with an MHE estimator. With simulation experiments on a two-stage HC MPprocess, it can be shown that state variables can be inferred reliably from limited measurement data, and thatm-econ MPC achieves significant improvements in economic performance.This chapter is outlined as follows. A brief introduction of full information estimation and MHE is givenin Section 6.2. Section 6.3 is devoted to the application of MHE in the MP process. The combination ofMHE and m-econ MPC is elaborated in Section 6.4. In Section 6.5, we present a simulation example todemonstrate the performance of the proposed m-econ MPC technique and MHE framework, followed by asummary of this chapter in Section 6.6.6.2 State Estimator Design6.2.1 Full Information EstimatorGiven the system model in (2.11a)–(2.11b) with the constraints on the state xk ∈ X and on the disturbanceζk ∈W. We assume the constraint sets X andW are closed with 0 ∈W.We first formulate the optimal constrained estimation as the solution to the following optimization prob-lem,J∗t = minz0,{ζk}t−1k=0JFIE(z0,{ζk}t−1k=0) (6.1a)s.t. zk+1 = f (zk,uk,ζk), k = 0, · · · , t−1, (6.1b)yk = g(zk)+ηk, k = 0, · · · , t, (6.1c)zk ∈ X, k = 0, · · · , t, (6.1d)ζk ∈W, k = 0, . . . , t−1, (6.1e)with the objective (6.1a) in the following form,JFIE(z0,{ζk}t−1k=0) :=t−1∑k=0L(ζk,ηk)+Γ(z0), (6.2)where the stage cost L : Rnη ×Rnζ 7→ R≥0 for all k ≥ 0. The initial penalty Γ : Rnx 7→ R≥0 summarizes the626.2. State Estimator Designprior information and satisfies Γ(x¯0) = 0, where x¯0 is the most likely value of z0, and Γ(zk) > 0, ∀zk 6= x¯0.The solution to (6.1a)–(6.1e) is the estimated state zˆ0 and disturbance {ζˆk}t−1k=0. Then the current state ofthe system zˆt can be calculated by forward programming given the estimated initial state zˆ0, the knowndeterministic input signal {u0, · · · ,ut−1}, and the estimated noise {ζˆk}t−1k=0.The state estimation problem formulated in (6.1a)–(6.1e) is known as the full information estimator(FIE). The derivation of FIE is based on the maximization of the a-posteriori Baysian estimation. In FIE, allthe past available measurements are used for the estimation. Solving FIE problem, however, is computationaldemanding since the problem dimension grows unbounded with time t as more data is processed. Especially,when the system dynamic is nonlinear and constrained, the online solution to the FIE problem is impracticaldue to the increasing computational burden. MHE strategy is then proposed to reduce the FIE problemto a fixed dimension optimization problem, in which the past data that is not considered any more will beapproximated by arrival cost so that the influence of the past information is still considered.6.2.2 Moving Horizon EstimatorMHE uses an online optimization as a means for designing the state estimator. Nonlinear MHE is an op-timization based strategy for state estimation for systems with nonlinear models and inequality constraints.Rather than determining the full state sequence as FIE, MHE estimates the truncated sequence {zˆk}tk=t−Nmheand {ζˆk}t−1k=t−Nmhe , where Nmhe is referred to as estimation horizon length which is the size of past date takeninto account.Consider the FIE in (6.1a)–(6.1e), the original objective function (6.2) can be rearranged and brokeninto two terms with time intervals t1 = {k : 0≤ k ≤ t−Nmhe−1} and t2 = {k : t−Nmhe ≤ k ≤ t−1} as thefollowing,Jmhe(z0,{ζk}t−1k=0) =t−1∑k=0L(ζk,ηk)+Γ(z0),=t−Nmhe−1∑k=0L(ζk,ηk)+t−1∑k=t−NmheL(ζk,ηk)+Γ(z0) (6.3)Thus, MHE can be formulated as follows,minzt−Nmhe ,{ζk}t−1k=t−Nmhet−1∑k=t−NmheL(ζk,ηk)+Ξt−Nmhe(zt−Nmhe), (6.4)636.2. State Estimator DesignwhereΞt−Nmhe(p) = minz0,{ζk}t−Nmhe−1k=0{Jmhe(x0,{ζk}t−Nmhe−1k=0 ) : zt−Nmhe = p}, (6.5)which subjective to the constraintsxk ∈ X, k = 0, · · · , t−Nmhe, and ζk ∈W, k = 0, · · · , t−Nmhe−1. (6.6)Remark 6.2.1. In the case where t ≤ Nmhe, FIE (6.1a)–(6.1e) is utilized. No data will be discarded at thispoint until the estimation horizon window is filled up with measurement data. After there is enough data inthe estimation window, MHE will be in action.Remark 6.2.2. In the MHE formulation (6.4)–(6.6), the initial penalty Ξt−Nmhe is referred to as arrival cost.The arrival cost is one of the most important components in MHE since it compresses the process dataand summarizes the effect of the past measurement data {yk}t−Nmhek=0 , which are not included in the finiteestimation window, on the state xt−Nmhe . In most linear systems, the arrival cost is chose to be the penalizeddeviation of xt−Nmhe away from the current estimate x¯t−Nmhe|t−Nmhe−1. However, when the system is nonlinearor constrained, it is difficult to derive a general algebraic expression for or to under-bound the arrival cost.Remark 6.2.3. Observability is one of the most importance concepts in the state estimator design. A systemis said to be observable if it is possible to uniquely infer the process state from the input-output data. Forlinear systems, the observability rank condition is applied, i.e., to check if the rank of the observabilitymatrix is equal to the state dimension. However, unlike linear system theory, there does not exist a widelyaccepted condition to check the observability for nonlinear systems since the observability of nonlinearsystem may depend on the system input, i.e. different inputs may lead to different state trajectories. As aresult it is impossible to uniquely reconstruct the state. For nonlinear systems, detectability or incrementalobservability, which are much weaker conditions than observability, is used.Definition 6.2.1. A system is said to be detectable if its unobservable states are asymptotically stable.Definition 6.2.2. [73] A system is incrementally observable if there exists a positive integer N>0 and aKfunction φ such that for any two state x1 and x2 the following condition stands,φ(||x1− x2||)≤k−1∑j=0||y( j,x1)− y( j,x2)||, ∀k ≥ N>0.646.3. MHE Design for a Two-stage HC MP ProcessThe incrementally observable condition states that if the prediction residuals are small for a sufficient periodof time, then the estimation error is small.Remark 6.2.4. In this thesis, formal observability analysis is not performed. We assume that the nonlineartwo-stage HC MP process is observable or detectable from a technical perspective.6.3 MHE Design for a Two-stage HC MP ProcessThe mathematical models of the two-stage HC MP process have been presented in Chapter 2.3. In previouschapters, we develop different advanced nonlinear econ MPC techniques assuming that all the measurementsare available at each sampling time. However, measurements of the MP system are normally not all availableat the same time or same rate. In this section, we develop an MHE for the nonlinear two-stage HC MPprocess.Given the two-stage HC MP process defined in (2.11a)–(2.11b), the measurement vector yt is defined asyt =[Pˇ,Mˇp,Mˇs]. (6.7)where Pˇ, Mˇp, Mˇs are the noisy measurements of the state variables P, Mp, Ms, respectively.For an effective control, MPC has to account for changes in state variables and therefore unmeasurablestates have to be estimated to solve the related optimization problem. Thus, the unmeasured states needto be inferred from the available (mostly noisy) measurements. In a two-stage HC MP process, the model(2.11a)–(2.11b) is normally corrupted with process and measurement noise ζt and ηt . Here we assume thatthe process disturbance and measurements are normally distributed with zero mean and constant covariancePζ and Pη :ζt ∼ N(0,Pζ ), ηt ∼ N(0,Pη). (6.8)The state variables need to be inferred from the measured outputs (6.7). Specifically, MHE uses a movingmeasurement window of the form:ITt = [IytT, IutT ] = [yt−Nmhe , · · · ,yt−1,yt ,ut−Nmhe , · · · ,ut−1], t ≥ 0, (6.9)to compute the estimates xˆt of the states xt . It is the information vector at time t. For the MP process656.4. Integration of MHE and M-econ MPC for MP Processesdiscussed in this chapter, yt is defined in (6.7). We formulate the estimation problem as the solution to thefollowing optimization problem,minz0,{ζk}Nmhe−1k=0Jmhe := µ||z0− x¯0t−Nmhe ||2+Nmhe∑k=0||yt−Nmhe+k−g(zk)||2Qη +Nmhe∑k=0||ζt−Nmhe+k||2Qζ (6.10a)s.t. zk+1 = f (zk,ut−Nmhe+k,ζk), k = 0, . . . ,Nmhe−1, (6.10b)yk = g(zk)+ηk, k = 0, . . . ,Nmhe, (6.10c)zk ∈ X, zNmhe ∈ X f , k = 0, . . . ,Nmhe, (6.10d)where Jmhe is the optimal cost which incorporates the arrival cost (the first term in (6.10a)) and the least-square error of the outputs (the second term in (6.10a)) over the estimation horizon Nmhe. x¯0t−Nmhe is the priorvalue of the initial state and µ is a weighting factor for the arrival cost. The state variables are subject to thestate model as well as the constraints in (6.10d). The solution of the optimization problem is given by thestate trajectory {zˆ0, · · · , zˆNmhe}. From the solution, we obtain the estimate of the current state of the system asxˆt ← zˆNmhe . Conditions for stability of MHE have been established for general settings in [1, 71, 72, 81, 102].6.4 Integration of MHE and M-econ MPC for MP Processes6.4.1 Basic NotationIn what follows, we assume that the tr MPC is feasible for any xt ∈ X. The optimal trajectory at time t isdefined as {x˜k|t , u˜k|t}Nmpc−1k=0 and is obtained by solving the optimization problem with the estimated state xˆt ,minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0Ltr(zk|t ,vk|t), (6.11a)s.t. z0|t = xˆt , zNmpc|t ∈ X f , (6.11b)zk+1|t = f (zk|t ,vk|t ,ζk|t), k = 0, . . . ,Nmpc−1, (6.11c)yk|t = g(zk|t)+ηk|t , k = 1, . . . ,Nmpc, (6.11d)zk|t ∈ X, k = 1, . . . ,Nmpc−1, (6.11e)vk|t ∈ U, k = 0, . . . ,Nmpc−1. (6.11f)666.4. Integration of MHE and M-econ MPC for MP ProcessesFor the nonlinear MP process defined in (2.11a)–(2.11b), we require that the terminal states lie in the terminalregion X f instead of at the desired steady state xss. X f ∈ X is a compact terminal region containing aneighborhood of the point xss in its interior. The value function for tr MPC at time t+1 is shown to beV˜ trt+1 :=Nmpc−1∑k=0Ltr(x˜k|t+1, u˜k|t+1). (6.12)In the standard tr MPC problem, the stage cost is normally defined as penalizing the deviation of processvariables from their steady-state values. The closed-loop stability of tr MPC is established by treating theoptimal cost along the closed-loop trajectory as a Lyapunov function [8, 31, 47, 68]. However, for econ MPC,the corresponding stage cost function is defined by Lec(·), which may be selected by the user according topractical needs and such function may not be positive definite. As a result, most practical economic costfunctions cannot be used as Lyapunov functions and closed-loop stability cannot be guaranteed. In the nextsubsection, we combine tr MPC and econ MPC into the m-econ MPC formulation to enforce stability.Imposing a tr MPC as a stabilizing constraint to the econ MPC results in the following m-econ MPCformulation similar as the one in Chapter 4. Note that the difference of the m-econ MPC formulation in thischapter is that we use the estimated state xˆt from MHE as the initial condition rather than xt as in (5.4a)–(5.4f),minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0Lec(zk|t ,vk|t), (6.13a)s.t. z0|t = xˆt , zNmpc|t ∈ X f , (6.13b)zk+1|t = f (zk|t ,vk|t ,ζk|t), k = 0, . . . ,Nmpc−1, (6.13c)yk|t = g(zk|t)+ηk|t , k = 1, . . . ,Nmpc, (6.13d)zk|t ∈ X, k = 1, . . . ,Nmpc−1, (6.13e)vk|t ∈ U, k = 0, . . . ,Nmpc−1, (6.13f)Nmpc−1∑k=0Ltr(zk|t+1,vk|t+1)≤ εt+1(σ), k = 0, . . . ,Nmpc−1, (6.13g)withεt+1(σ) := V˜ trt+1+σ(Vtrt −V˜ trt+1). (6.14)where σ ∈ [0,1) is the tuning parameter.676.4. Integration of MHE and M-econ MPC for MP Processes6.4.2 MHE and M-econ MPC Framework for MP ProcessesThe MP process is a complex MIMO nonlinear process with strong interactions among variables. Basedon the two-stage HC MP process model developed in Chapter 2, the m-econ MPC developed in Chapter 5,and MHE approaches presented in the previous section, we are now in a position to combine the MHE andm-econ MPC for MP processes. The graphical depiction of the integrated m-econ MPC and MHE frameworkis shown in Figure 6.1.ControllerMP process𝒙𝒕+𝟏 = 𝒇(𝒙𝒕, 𝒖𝒕, 𝜻𝒕)Sensor𝒚𝒕 = 𝒈 𝒙𝒕 + 𝜼𝒕EstimatorSetpointProcess uncertaintyMeasurement noise𝒖𝒕𝜻𝒕𝜼𝒕𝒚𝒕ෝ𝒙𝒕Figure 6.1: Integrated controller and estimator for MP processIn the m-econ MPC and MHE framework design for the MP process, the tracking and economic objec-tive functions are similar as those defined in (3.6). The difference in the following formulation is that theestimated states xˆk|t are considered in the objective functions,V trt =Nmpc−1∑k=0||xˆk|t − xss||2Qx + ||uk|t −uss||2Qu , (6.15)V ect =Nmpc−1∑k=0T SEk|t , (6.16)where Qx, Qu are positive-definite weighting matrices for the state and input variables, respectively. T SEkis the total energy as defined in (2.2) and given by ([0,1,0,0,0] xˆk +[0,0,0,1,0] xˆk) / [1,0,0,0,0] xˆk for thespecific MP process under study.686.5. Simulation ResultsThe m-econ MPC optimization problem with MHE for the MP process can be formulated as follows,minv0|t ,··· ,vNmpc−1|tNmpc−1∑k=0T SEk|t , (6.17a)s.t. z0|t = xˆt , zNmpc|t ∈ X f , (6.17b)zk+1|t = f (zk|t ,vk|t ,ζk|t), k = 0, . . . ,Nmpc−1, (6.17c)xmin ≤ zk|t ≤ xmax, k = 1, . . . ,Nmpc−1, (6.17d)umin ≤ vk|t ≤ umax, k = 0, . . . ,Nmpc−1, (6.17e)Nmpc−1∑k=0||zk|t+1− xss||2Qx + ||vk|t+1−uss||2Qu ≤ εt+1(σ), (6.17f)withεt+1(σ) := V˜ trt+1+σ(Vtrt −V˜ trt+1). (6.18)where σ ∈ [0,1). The state estimate xˆt is calculated by solving MHE problem formulated in (6.10a)–(6.10d).Here, we note that the stabilizing constraint takes the form of a ball centered around the equilibrium point(xss,uss) and with radius εt+1(σ). Consequently, the stabilizing constraint can be interpreted as a trust-regionthat the MPC controller can visit to optimize economics while preserving stability.At each time instant, the optimal input sequence can be obtained by solving the m-econ MPC (6.17a)–(6.17f). Only the first input of the optimal input sequence will be injected to the MP plant. Given the newestmeasurements and past Nmhe-step of input and measurement data, the constraint MHE (6.10a)–(6.10d) willbe incorporated to the system to eliminate the noises on measured states and estimate the unmeasured states.For the MHE, we only consider the case where the model uncertainty ζt and measurement noises ηt arenormally distributed with zero mean and constant covariance Pζ and Pη , respectively. The detailed algorithmof implementing the simultaneous m-econ MPC and MHE is provided in Table 6.1.6.5 Simulation ResultsIn the simulation example of Chapter 4 , we apply the proposed m-econ MPC algorithm with different valuesof σ ∈ [0,1) under the assumption that state variables are directly available from online sensors. From thatsimulation, we show that the proposed m-econ MPC can not only reduce the energy consumption, but alsoguarantee the closed-loop stability. The m-econ MPC algorithm also allows the user to tune the controller in696.5. Simulation ResultsTable 6.1: The implementation of m-econ MPC and MHE for a two-stage HC MP processAlgorithm 6.1Input: x0 ∈ X, σ ∈ [0,1), set t← 0 and ε0(σ)←+∞.Loop: for t = 0, . . . ,T (simulation ends) do1: Solve the m-econ MPC optimization in (6.17a)–(6.17f) and (6.18) forthe state xt and εt(σ), evaluate V trt , and set ut ← v¯0|t .2: Implement ut to the plant and obtain the state variables xt+1 = f (xt ,ut ,ζt).3: Solve tr MPC in (6.11a)–(6.11f) for the state xt+1, and evaluate V˜ trt+1.5: Solve MHE in (6.10a)–(6.10d) to get the estimates [zˆ0, · · · , zˆNmhe ].6: Set εt+1(σ)← V˜ trt+1+σ(V trt −V˜ trt+1) and send the currentestimate zˆNmhe to m-econ MPC, i.e. xˆt+1← zˆNmhe .7: end loopsuch a way that a desired trade-off between the economic and tracking performances is achieved based onpractical demands. Based on the previous simulation results, in this example we demonstrate the effective-ness and economical benefits of using the m-econ MPC algorithm and MHE in a two-stage HC MP process.As discussed in Chapter 5, by applying m-econ MPC in a two-stage HC MP process about 10% to 27% ofSE reduction can be achieved. Thus, in this example we provide results for the simultaneous implementationof state estimation and m-econ MPC with a fixed tuning parameter σ . Moreover, the measurement noise andmodel uncertainty are considered in the this example. In the closed-loop simulation, the variations in rawmaterials such as the chip bulk density dc and the chip solid content sc are considered as the disturbancesand given in Chapter 5 (see Table 5.2).In this example, the MHE is incorporated in the two-stage HC MP system as part of the feedback con-trol of the m-econ MPC. The model uncertainty and measurement noise are considered in the closed-loopsimulation. The measured (but noisy) state variables in the HC MP process are production rate, primarymotor load and secondary motor load. The unmeasured state variables, which are the consistencies for theprimary and secondary refiners, are estimated by MHE from the noisy measurements and past input data. Inthis example, the simulation duration is T = 450 samples. The prediction and control horizons for m-econMPC are selected to be equal and set to be Nmpc = 30. The sampling interval is 2s. The weighting matri-ces Qx = diag{[0.01,10,0.1,10,0.1]}, and Qu = diag{[0.1,100,0.01,100,0.01]}. The estimation horizonNmhe = 15 with arrival cost weighting vector selected to be µ = [0.5,1,0.1]. It is assumed that the simulationand real MP process share the same disturbance and measurement noise covariance Pζ = [1,0.1,1,0.1,0.5],Pη = [0.1,0.1,0.1,0.1,0.1], respectively. The tuning parameter σ is fixed and chosen to be 0.8. The otherparameters are set the same as those in the previous simulation.706.6. SummaryThe simulation results are shown in Figure 6.2–6.4. The manipulated variables for the closed-loop two-stage HC MP process are given in Figure 6.2. From Figure 6.2, we can find that all manipulated variablesin the two-stage HC MP process are able to track the setpoints within 50 samples. Besides, the outputvariables, which are actually states x1, x2 and x3 (plus noise), can approach the respective setpoints quicklyunder the proposed integrated MHE and m-econ MPC framework. The tracking performance of all fivestates are illustrated in Figure 6.3. It is apparent from Figure 6.3 that MHE can yield precise estimates forboth measured and unmeasured states, which is evidenced by the large overlapping between the actual andestimates states. The SE consumption in this example is shown in Figure 6.4, and the energy saving rate withσ = 0.8 is about 30% compared with tr MPC.6.6 SummaryThis chapter presents an integrated framework consisting of m-econ MPC and MHE for a nonlinear two-stageHC MP process. The m-econ MPC inherits the merits of economic MPC in reducing energy consumptionand that of tr MPC in ensuring the closed-loop stability. It is shown that the m-econ MPC enables the user tomake an informed trade-off between production economics and the convergence speed of process variablesto the setpoints. For industrial processes with unmeasured states, including the MP processes, we propose tocombine online MHE together with the m-econ MPC into an integrated framework that can be implementedin practice. Simulation results from MP processes are provided to demonstrate the advantages of usingm-econ MPC and the effectiveness of the proposed scheme that combines the m-econ MPC and MHE.716.6. Summary50 100 150 200 250 300 350 400 450Screw speed(RPM)02040Setpoint Manipulated input50 100 150 200 250 300 350 400 450Prim gap(mm)02450 100 150 200 250 300 350 400 450Prim flow rate(L/min)050050 100 150 200 250 300 350 400 450Sec gap(mm)024Time in Samples50 100 150 200 250 300 350 400 450Sec flow rate(L/min)0100200Figure 6.2: The manipulated variables of the MP process726.6. Summary100 150 200 250 300 350 400 450Production rate(tonnes/day)200400600Setpoint Actual state Estimated state50 100 150 200 250 300 350 400 450Prim ML(MW)051050 100 150 200 250 300 350 400 450Prim cosistency(%)20406050 100 150 200 250 300 350 400 450Sec ML(MW)0510Time in Samples50 100 150 200 250 300 350 400 450Sec cosistency(%)304050Figure 6.3: The state variables of the MP process736.6. SummaryTime in Samples50 100 150 200 250 300 350 400 450Specific energy(kWh/t)020040060080010001200Figure 6.4: SE74Chapter 7A Tractable Approximation for StochasticModel Predictive Control and Application toMP Process7.1 IntroductionOver the past few decades, significant efforts have been made towards a better control and optimizationscheme for the MP process from both the academia and industry [3, 4, 35, 62, 66, 94]. Currently, determin-istic models are dominantly used for modeling MP processes and designing control polices [43, 44, 92–94].Despite the benefits of fast calculation, deterministic modeling approaches are unable to explicitly take intoaccount the stochastic uncertainties and disturbances.In this work, we explore tractable approximations for SMPC. Under the proposed paradigm, we usesample scenarios to transform statistical measures for the objective and constraints into finite-dimensionalrepresentations [15, 76, 87]. To deal with tractability issues of the resulting optimization problems, wepropose an approximation technique that is inspired by the quantile scenario analysis method proposed in[100]. In this approach, we solve multiple DMPC problems for different scenarios to obtain a set of candidatecontrol policies. The observation is that these policies can be computed quickly and in parallel as they do notinvolve statistical measures. The set of computed control policies forms a candidate set from which we selectthe policy that best approximates the SMPC solution (i.e., yields the smallest statistical measures for theobjectives and constraints). This approach allows us to handle complex measures and allows us to prioritizeconflicting objectives such as economics and stability. The proposed approach only provides an approximatepolicy of the SMPC problem but we note that this approach can be interpreted as a controller that seeks tofind the DMPC policy that best approximates the performance of SMPC. Moreover, since MPC policy is757.2. Stochastic Model Predictive Controlequivalent to solving a problem with one scenario (typically the mean or worst case), the proposed approachcan do no worse than the standard DMPC policy. We demonstrate the developments using a stochasticversion of econ MPC [103] applied to a MP process. In this process, we seek to drive transitions betweensteady-states that deliver product with desired characteristics and while minimizing energy consumption [44–46]. The process is challenging in that it involves multiple sources of uncertainty and strong nonlinearities.The proposed framework extends the work in deterministic econ MPC for MP processes we developed inChapter 5 – 6.This chapter is organized as follows. In Section 7.2, we provide a discussion of SMPC. Section 7.3 isdevoted to the proposed approximation strategy. A simulation case study to demonstrate the performance ofthe proposed approach to the two-stage HC MP process is given in Section 7.4 followed by conclusions inSection 7.5.7.2 Stochastic Model Predictive ControlGiven the two-stage HC MP process model defined in (2.11a)–(2.11b) with the manipulated and state vari-ables listed in Table 2.1. Let ζ (S)t := {ζ 10|t , . . . ,ζ SNmpc−1|t} and η(S)t := {η11|t , . . . ,ηSNmpc|t} be the sets of i.i.d.samples of ζt and ηt , respectively, drawn at time t, where Nmpc is the prediction horizon. For convenience, ascenario set is defined as {ζ (S)t ,η(S)t } ∈Ω, where S is the number of samples or scenarios. The i-th scenarioof ζ ik|t , k = 0, . . . ,Nmpc−1, and η ik|t , k = 1, . . . ,Nmpc is denoted as {ζ ik|t ,η ik|t}Nmpc−1k=0 , i ∈ {1, . . . ,S}, in the restof this chapter. Then for the general process model described in (2.11a)–(2.11b), the SMPC has the form:minv0|t ,··· ,vNmpc−1|t1SS∑i=1Nmpc−1∑k=0Lec(zik|t ,vk|t), (7.1a)s.t. zik+1|t = f (zik|t ,vk|t ,ζik|t), k = 0, · · · ,Nmpc−1, (7.1b)yik|t = g(zik|t)+ηik|t , k = 1, · · · ,Nmpc, (7.1c)zi0|t = xt , ziNmpc|t ∈ X f , (7.1d)zik|t ∈ X, k = 1, · · · ,Nmpc−1, (7.1e)vk|t ∈ U, k = 0, · · · ,Nmpc−1, (7.1f)N−1∑k=0Ltr(zik|t ,vk|t)≤ εt , k = 0, . . . ,Nmpc−1, (7.1g)767.2. Stochastic Model Predictive Controlwhere the stage cost is given by Lec(zik|t ,vk|t). zik|t , yik|y denote the k-step-ahead predictions of state andoutput variables at time t for the i-th scenario. With the scenario set {ζ (S)t ,η(S)t }, the system dynamic (7.1b)provides S different state trajectories over the prediction horizon, each corresponding to a particular scenario{ζ ik|t ,η ik|t}Nmpc−1k=0 , i ∈ {1, · · · ,S}. The cost function is averaged over all S scenarios. The state measurementat the current sampling time t (i.e., xt) is used as initial state. For convenience, we define the economic valuefunction evaluated at scenario i and under input vk|t as:V ect (vk|t ,{ζ ik|t ,η ik|t}) :=Nmpc−1∑k=0Lec(zik|t ,vk|t) (7.2)The SMPC problem seeks to find the optimal policy {v¯0|t , · · · , v¯Nmpc−1|t} that minimizes the expectedcost and satisfies constraints (7.1a)–(7.1g). Only the first element of the policy ut := v¯0|t is injected into theplant. The constraints of the SMPC formulation include standard input and state constraints and a stabilizingconstraint. To construct the stabilizing constraint, we assume that the stage cost Ltr is a positive definitefunction (e.g., as in a tracking problem) and consider a sequence {εt} that decreases as t → ∞. Detailson how to construct such a sequence can be found in [41, 103] (details are omitted here for brevity). Forconvenience, we also define the tracking value function at scenario i and input vk|t as:V trt (vk|t ,{ζ ik|t ,η ik|t}) :=Nmpc−1∑k=0Ltr(zik|t ,vk|t) (7.3)We thus see that the stabilizing constraint seeks to progressively decrease in the tracking function in orderto ensure stability [31, 92, 103]. This constraint is necessary because minimization of the economic costfunction does not guarantee stability. In Chapters 5 and 6, we developed a deterministic variant of thiseconomic MPC formulation. The formulation considered here is a stochastic variant of such formulation.In summary, the idea behind SMPC controller is to compute an optimal control policy that minimizes theexpected economic cost (corresponding to using the expected value as the statistical measure). An issue withthe use of the expected value as a measure is that it might lead to poor performance in extreme scenarios.The proposed SMPC formulation can thus be modified to by using alternative statistical measures such asthe quantile or the median (quantile at a probability of 50%) of the economic cost (this is a more robustapproach to deal with extreme scenarios). The control policy computed with SMPC must also be feasibleunder all S scenarios. In other words, the SMPC formulation enforces satisfaction of state constraints and777.2. Stochastic Model Predictive Controldecrease of the tracking value function for all scenarios S. We note that assuming that the constraints holdfor all scenarios is equivalent to say that they hold with probability one:P(Nmpc−1∑k=0Ltr(zk|t ,vk|t)≤ εt)= 1. (7.4)Consequently, this formulation might be restrictive. The proposed SMPC formulation can thus be modifiedto enforce constraints by using alternative measures such as a chance constraint in which the constraintsare enforced with a probability lower than one. In addition, we can relax the satisfaction of the stabilizingconstraint for all scenarios by requiring satisfaction of a probability ρ < 1:P(Nmpc−1∑k=0Ltr(zk|t ,vk|t)≤ εt)≥ ρ (7.5)or by requiring satisfaction in expectation:1SS∑i=1Nmpc−1∑k=0Ltr(zik|t ,vk|t)≤ εt (7.6)While the SMPC formulation makes practical sense, it can be challenging to solve in real-time. This can bedue to the need to handle many scenarios and/or from the need to capture complex measures (e.g., quantilesand chance constraints). Consequently, we are interested in developing strategies that compute approximatecontrol policy.787.3. Approximating the Policy of SMPC7.3 Approximating the Policy of SMPCIn this chapter, we propose a strategy to compute an approximate policy for SMPC. For each scenario i =1, ...,S, an optimal policy {v¯i0|t , . . . , v¯iNmpc−1|t} is computed by solving a DMPC problem of the form:minvi0|t ,··· ,vNmpc−1|tNmpc−1∑k=0Lec(zk|t ,vk|t), (7.7a)s.t. z0|t = xt , zNmpc|t ∈ X f , (7.7b)zik+1|t = f (zk|t ,vk|t ,ζk|t), k = 0, . . . ,Nmpc−1, (7.7c)yik|t = g(zik|t)+ηk|t , k = 1, . . . ,Nmpc, (7.7d)zik|t ∈ X, vk|t ∈ U, k = 0, . . . ,Nmpc−1, (7.7e)Nmpc−1∑k=0Ltr(zk|t+1,vk|t+1)≤ εt , k = 0, . . . ,Nmpc−1. (7.7f)The corresponding value function for the i-th scenario is given by V ect (u¯i0|t ,{ζ ik|t ,η ik|t}). We define the policycandidate set over the S scenarios as u¯0 = {u¯10|t , · · · , u¯S0|t}.Given the optimal policy u¯i0|t for the i-th scenario, we evaluate the value functions Vect (u¯i0|t ,{ζ jk|t ,η jk|t})using this policy over for the full set of scenarios j 6= i. Our goal now is to select a control policy from thecandidate set to be implemented in the system. This selected policy must solve (or approximately solve)the SMPC problem. In the context of problem (7.7a)–(7.7f), we want a control policy that minimizes theexpected cost and satisfies the state and stabilizing constraints for all scenarios (input constraints are satisfiedby construction).To select a policy, we construct a coordinating matrix for the cost (denoted as V ecset and shown in (7.8));here; each row corresponds to a candidate control and each column to a scenario. We note that the diagonalelements correspond to the optimal cost of problem (7.7a)–(7.7f) for all scenarios i.V ecset :={ζ 1k|t ,η1k|t} {ζ 2k|t ,η2k|t} · · · {ζ Sk|t ,ηSk|t}u¯10|tu¯20|t...u¯S0|tV¯ ect (u¯10|t ,{ζ 1k|t ,η1k|t})V ect (u¯20|t ,{ζ 1k|t ,η1k|t})...V ect (u¯S0|t ,{ζ 1k|t ,η1k|t})V ect (u¯10|t ,{ζ 2k|t ,η2k|t})V¯ ect (u¯20|t ,{ζ 2k|t ,η2k|t})...V ect (u¯S0|t ,{ζ 2k|t ,η2k|t}))· · ·· · ·. . .· · ·V ect (u¯10|t ,{ζ Sk|t ,ηSk|t})V ect (u¯20|t ,{ζ Sk|t ,ηSk|t})...V¯ ect (u¯S0|t ,{ζ Sk|t ,ηSk|t})(7.8)797.3. Approximating the Policy of SMPCIn an ideal case, the state and stabilizing constraints are satisfied for all scenarios j 6= i and for allcandidate inputs u¯i0|t . In such a case, our strategy simply selects the input that leads to the smallest expectedcost (column-wise average). This approximation approach is fast because it only requires solving decoupledscenario problems (which can be done individually and in parallel). Notably, this approach can also be usedto compute actions that minimize alternative statistical measures for the cost. For instance, we can also selectthe control candidate that achieves the smallest quantile of the cost [100]. This can be done trivially usinginformation from the cost matrix but doing so directly in an SMPC formulation is non-trivial (quantiles donot have a simple algebraic forms as in the case of expected values). Our approach thus seeks to not onlyfind approximate policies faster but also to enable the use of alternative statistical measures in SMPC.The proposed approximation approach presents interesting properties. Specifically, we observe that thesample average of the diagonal entries in V ecset provides a lower bound for the optimal cost of SMPC. In thestochastic programming literature, this lower bound is the so-called wait-and-see cost [14]. Consequently,it is possible to estimate an optimality gap for every candidate control (estimate how far is the approximatepolicy from the actual SMPC policy).We also highlight that the candidate controls considered are not constructed arbitrarily but are built byexploring the actual uncertainty space. As a result, we expected that the optimal SMPC policy is in a spacespanned by the candidate controls (or at least close to that space). In fact, we highlight that some of thecandidate control policies correspond to policies computed under typical DMPC formulations. Specifically,in a DMPC formulation, one often selects a single representative scenario (S = 1) for the random variables(typically the mean or the worst-case) to compute the control. Consequently, we observe that the policyobtained with the proposed approximation approach can do no worse than a typical DMPC policy. Ourapproach can also be interpreted as a strategy that seeks to improve the DMPC policy or as a strategy thatseeks to find a deterministic policy that best approximates the SMPC policy. The detailed approximationscheme is outlined in Table 7.1.In the ideal case, we assume that the state and stabilizing constraints hold for all scenarios and all can-didate controls. In other words, it is assumed that a single input can satisfy all constraints. However, inpractice, we expect this assumption not to hold. In fact, it is possible that the SMPC formulation does noteven have a feasible solution. Because of this, one must select a control that allows for some constraintviolations. One possibility to deal with this is the following: given scenarios {ζ jk|t ,η jk|t}Nmpc−1k=0 , j = 1, . . . ,S,and candidate control policies, we construct matrix V ecset as in the ideal case. For a given candidate policy807.3. Approximating the Policy of SMPCTable 7.1: Implementation of proposed approximate SMPC scheme in ideal caseAlgorithm for the ideal caseInput: x0 ∈ X, σ ∈ [0,1), set t← 0 and ε0←+∞.For t = 0, . . . , simulation duration do1: Draw S scenarios of {ζ ik|t ,η ik|t}Nmpc−1k=0 , i = 1, . . . ,S.2: for i = 1, . . . ,S do2.1. Compute optimal input u¯i0|t and its corresponding optimal value functionV¯ ect (u¯i0|t ,{ζ ik|t ,η ik|t}) for the i-th scenario.2.2. Given u¯i0|t computed in step 2.1, evaluate Vect (u¯i0|t ,{ζ jk|t ,η jk|t})for the rest of scenarios {ζ jk|t ,η jk|t}Nmpc−1k=0 , ∀ j = 1, · · · ,S, and j 6= i.end for3: Construct matrix V ecset as structured in (7.8).4: Find the minimal measure in each row of V ecset and its corresponding input u¯l0|t ,l ∈ 1, · · · ,S. Set ut ← u¯l0|t .5: Implement ut to the plant and obtain the state variables xt+1.6: Set εt+1 < εt .Endu¯i0|t , i ∈ {1, · · · ,S}, we count the number of state constraint (7.7e) violations (denoted as Mi) and count thenumber of stabilizing constraint (7.7f) violations (denoted as Ni) over all scenarios j = 1, . . . ,S. We thensingle-out policies u¯i0|t ∈ u¯0 that have the smallest faction of violations Ni/S, i ∈ {1, . . . ,S} and that satisfyMi/S≤ ρ (where ρ ∈ [0,1] is a given probability level). By construction, we have that Ni ≤ S and Mi ≤ S andwe thus note that minimizing Ni/S is equivalent to minimizing the probability (frequency) of state constraintviolations and the requirement Mi/S≤ ρ is a sample approximation of a chance constraint and ρ is a desiredprobability level [24]. Specifically, when ρ = 0, no violations of the stabilizing constraint are allowed and,when ρ = 1, all stabilizing constraints are allowed to be violated. We then construct a reduced set of can-didate policies u¯1 = {u¯i0|t , . . .} ⊂ u¯0 that meet the constraint satisfaction criteria. To choose the best overallpolicy, the coordinate matrix V ecset is reduced into Vˆecset to account only for values that satisfy the constraintcriteria. The policy u¯0|t to be implemented is thus the one leading to the smallest cost measure (e.g., theexpected value or median) among all columns in Vˆ ecset . In this general case, we assume that at least one ofthe optimal inputs meets the constraint satisfaction criteria. The detailed approximation algorithm for thisgeneral case is outlined in Table 7.2. In the ideal case, this algorithm simply reduces to that in Table 7.1because the matrices V ecset and Vˆecset coincide.The proposed approach is an approximate strategy for SMPC but has several practical benefits. First ofall, the approach is intuitive and easy to explain to industrial practitioners. Moreover, the approach is flexible817.4. Application to Mechanical Pulping Processesin that it can handle different statistical measures and enables prioritization of constraints. A key observationand theoretical justification of the approach is that it can do no worse than DMPC. This is because thecontrol policy of DMPC is one of the candidate policies. Specifically, in DMPC, a representative valuefor the uncertainties (e.g., the mean or the worst-case value) is chosen to compute the control policy. Theproposed approach can thus be interpreted as a strategy that seeks to improve on the DMPC policy or as astrategy that seeks to find the DMPC policy that best approximates the stochastic policy.7.4 Application to Mechanical Pulping ProcessesWe now demonstrate the effectiveness of the proposed algorithm. The MP process model is given in (2.11a)–(2.11b). The process dynamics are modeled through system identification with real industrial data. In thesimulations, we assume that all the state variables are directly measurable and affected by random measure-ment noise (denoted as ηt). We also introduce random disturbances to the properties of the wood chips(denoted as ζt). To enforce stability, we obtain a factor εt by solving a reference tracking problem, as re-ported in [92]. We implemented the MP process in AMPL and solved the nonlinear optimization problemsusing IPOPT [96].Variations in the raw materials such as the chip bulk density and chip solid content are considered asrandom disturbances. We assume that the disturbances {ζt ,ηt} are normally distributed with zero mean andconstant covariance:ζt ∼N (0,Qζ ), and ηt ∼N (0,Qη).The prediction horizon is set to be Nmpc = 30. The sampling interval is chosen as 2 sec, and the simulationtime is 250 sec. The other parameters used in the simulation are shown in Table 7.3. To demonstrate theeffect of the number of scenarios S on control performance, we selected three cases with S = 1,5,30 and theSMPC controller is designed to minimize the median of the economic cost. When S = 1, it is assumed thatonly one sample is drawn (this approach is equivalent to a DMPC policy).Table 7.4 shows the variances for states, manipulated inputs, and SE over time. The variance is usedas a subject of volatility in the controller performance. It can be observed that by increasing S from 1 to30, the variance of measured output variables, manipulated input variables and the SE have been reduceddramatically. This is because using a single scenario makes the control policy susceptible to variations in thedisturbances while increasing the number of scenarios protects the controllers.827.5. SummaryThe simulation results are shown in Figures 7.1 – 7.3. Figures 7.1 and 7.2 illustrate the tracking per-formance of measured outputs and manipulated input variables. It can be observed that, in all cases, thecontroller is able to stabilize the system at the desired target steady-state. For S = 1, the controller gets closeto instability at the beginning of the transition while, as S increases, the controller has better performance.7.5 SummaryThe main contribution of this chapter is to develop an approach to obtain approximate control policies forSMPC. This approach seeks to address computational tractability issues of SMPC and offers flexibility tohandle diverse statistical measures. The proposed approach offers the guarantee that it can do no worsethan DMPC and can be interpreted as a strategy that seeks to find a deterministic policy that gives the bestapproximation to a SMPC policy. A simulation example for a two-stage HC MP process demonstrates theeffectiveness of the proposed approach.837.5. Summary0 50 100 150 200 250300400500600Production Ratetonnes/dayState VariablesSetpoint S=30 S=5 S=10 50 100 150 200 2500510Primary MLMW0 50 100 150 200 25036384042Primary Consistency%0 50 100 150 200 2500510Secondary MLMW0 50 100 150 200 250Time in Samples303540Secondary Consistency%Figure 7.1: Closed-loop output policies obtained with the proposed controller847.5. Summary0 50 100 150 200 25020304050Screw SpeedRPMManipulated Input VariablesSetpoint S=30 S=5 S=10 50 100 150 200 250024Primary Gapmm0 50 100 150 200 250300400500Primary FLRL/min0 50 100 150 200 250024Secondary Gapmm0 50 100 150 200 250Time in Samples0100200Secondary FLRL/minFigure 7.2: Closed-loop input policies obtained with the proposed controller857.5. Summary0 50 100 150 200 250Time in Samples020040060080010001200140016001800kWh/tSpecific EnergyS=30 S=5 S=1Figure 7.3: Comparison of the energy reduction achieved with the proposed controller867.5. SummaryTable 7.2: Implementation of proposed approximate SMPC scheme in general caseAlgorithm for general casesInput: x0 ∈ X, ρ ∈ [0,1), set t← 0 and ε0←+∞.For t = 0, . . . , simulation duration do1: Draw S scenarios of {ζ ik|t ,η ik|t}Nmpc−1k=0 , i = 1, . . . ,S.2: for i = 1, . . . ,S, do2.1. Compute optimal input u¯i0|t and its corresponding optimal value functionV¯ ect (u¯i0|t ,{ζ ik|t ,η ik|t}) for the i-th scenario.2.2. Given u¯i0|t computed in step 2.1, evaluate Vect (u¯i0|t ,{ζ jk|t ,η jk|t})for the rest scenarios {ζ jk|t ,η jk|t}Nmpc−1k=0 , ∀ j = 1, . . . ,S, and j 6= i.end for3: Combining the results computed in step 2, construct the optimal input set u¯0 ={u¯10|t , · · · , u¯S0|t} and the coordinate matrix V ecset as in (7.8).4: Check the constraints (7.7e) and (7.7f) and count the number of the violations Mi andNi, i = 1, · · · ,S, for each sampled scenarios {ζ jk|t ,η jk|t}Nmpc−1k=0 , j = 1, · · · ,S, underthe optimal manipulated input u¯i0|t , i 6= j.5: Find the inputs u¯i0|t ∈ u¯0 satisfying the conditions: (a) Mi/S≤ ρ; (b) have the leastnumber of Ni, i ∈ {1, · · · ,S}. Stack the qualified optimal inputs into a setu¯1 = {u¯i0|t , · · ·} ⊂ u¯06: Choose the best overall optimal input as follows:6.1. If there is only one candidate input in u¯1,then the best overall optimal input is chosen to be this candidate input u¯l0|t .6.2. If there are more than one candidate inputs in u¯1then6.2.1. A new matrix Vˆ ecset is formed with value functions for eachscenario under the qualified optimal inputs in u¯1;6.2.2. The best optimal input u¯l0|t corresponds to the row that contains theminimal measure among all rows in the new matrix Vˆ ecset .7: Set ut ← u¯l0|t . Implement ut to the plant and obtain the state variables xt+1.8: Set εt+1 < εt .EndTable 7.3: Simulation parameters for the SMPC controllerSymbol Values DescriptionT 250s Simulation lengthNmpc 30 Prediction horizonS {1, 5, 30} Number of the scenariosQw 0.1 Variance of the disturbance/model uncertaintyQv 0.1 Variance of the measurementρ 0.2 Violation tolerance for stabilizing constraint877.5. SummaryTable 7.4: Variances for outputs, inputs, and SE for scenario cases S = 1,5,30Variance of output variablesS = 1 [0.0505, 0.0546, 0.0394, 0.0406, 0.0468]S = 5 [0.0483, 0.0395, 0.0390, 0.0464, 0.0419]S = 30 [0.0342, 0.0432, 0.0314, 0.0351, 0.0336]Variance of input variablesS = 1 [0.0029, 0.0668, 63.2465, 0.1235, 96.7086]S = 5 [0.0023, 0.0491, 57.0844, 0.1337, 90.1703]S = 30 [0.0013, 0.0440, 34,4772, 0.0944, 69.2944]Variance of SES = 1 8.1378×104S = 5 6.6490×104S = 30 6.5866×10488Chapter 8Conclusions and Future Work8.1 ConclusionsTargeting at balancing the economic and tracking performance of the two-stage HC MP process, this thesisconsists of five aspects, namely nonlinear two-stage HC MP process models, econ MPC, m-econ MPC,nonlinear MHE, and the approximation of SMPC.First of all, we formulate a nonlinear model for the MIMO two-stage HC MP process. This mathemat-ical model is developed by combining mechanistic and empirical methods. The control and optimizationtechniques investigated in this thesis are all based on this model but with minor adjustment as needed in eachproblem formulations.As for the controllers, we firstly study the econ MPC. The econ MPC with two different penalizationterms are considered and compared in terms of control and optimization performance. To be specific, onepenalization term has penalty on the increment of the input (Scheme A) and the other on the offset of the inputfrom its steady-state (Scheme B). From the simulation results, it shows that the econ MPC with Schemes Aand B are able to ensure the pulp qualities within their bounds. Besides, the two schemes show a similarcontrol performance of the state variables, while Scheme B achieves a better input tracking performance.Moreover, extra SE reduction can be achieved by using Scheme A. However, for both Schemes A and B, ithas been observed that in order to reduce the energy consumption in terms of the SE, larger penalty has tobe added to the economic term in the objective function. As a result, it leads to a significant deviation of thestate variables from their steady-state target.Motivated by the trade-off observed in the econ MPC, the m-econ MPC is proposed as to interpretthe economic and tracking performance as two conflicting objectives in an econ MPC. In the proposed m-econ MPC, a standard tr MPC is exploited as a stabilizing constraint which preserves inherent stabilityof the auxiliary MPC controller. M-econ MPC can simultaneously minimize economic cost while enforcingsetpoint tracking. The scalar σ ∈ [0,1) is formulated in the proposed m-econ MPC that enables users to adjust898.1. Conclusionsthe trade-off between the economics and the convergence rate of the tracking performance. Specifically,when σ ∈ [0,1) decreases, the convergence speed of the states to their steady state setpoints are improved,whereas the economic saving drops. Especially, when σ = 0, the m-econ MPC is reduced to be a standardtr MPC. For the extreme case where σ = 1, the m-econ MPC will be equivalent to econ MPC withoutregulations. However, the convergence and stability cannot be guaranteed in this case since the target ismerely achieving the optimal economic performance regardless of the tracking performance or even thestability. In this sense, the tr MPC and econ MPC can be treated as two special cases of the m-econ MPC.The simulation results further demonstrate that about 10% to 27% of the SE can be saved by choosing the σvalue between 0.5 and 0.75.The aforementioned econ MPC and m-econ MPC techniques are based on the assumption that all thestate variables and pulp properties are directly measurable and known in real-time at each sampling time.However, due to the lack of fast and reliable measurement sensors, information on important variables suchas pulp consistency in HC refining is rarely available. To overcome these obstacles, we develop an MHEfor the nonlinear HC refining process, and propose a new m-econ MPC and MHE framework for the MPprocess. In this framework, the estimated state variables can be used directly in the m-econ MPC design ateach sampling time. The simulation results indicate a good control and estimation performance using ourproposed m-econ MPC and MHE framework. This framework makes it possible to apply the m-econ MPCtechniques in pulp mills where only limited online sensors are available.As an investigative work, we then develop an approximation technique for SMPC. In the literature,traditional MPC is usually based on the assumption that the process model, linear or nonlinear, is accurateand that future disturbances are constant. However, these assumptions are not valid in practice and canresult in poor closed-loop performance. In order to remove this assumption, in the proposed technique,the system dynamics are of a stochastic nature, and an additive measurement noise is introduced. In theproposed algorithm, the stochastic optimizations, which are difficult to solve, are decomposed into a set ofsimilar deterministic optimizations that can be efficiently solved using parallel computation. By solving theoptimization under each sampled scenario, which becomes a DMPC problem, the approximation methodselects the optimal input that yields the smallest median of cost function when evaluating this input acrossthe other scenarios. In this manner, the proposed method can be more robust to extreme scenarios that occurwith a low probability. Thus, the approximation SMPC is an alternative to existing SMPC techniques thatcannot be directly applied to nonlinear industrial processes, due to the excessive conservatism or numerical908.2. Future Workintractability caused by chance constraints.8.2 Future WorkAlthough we have made significant progress in developing advanced control and estimation techniques forthe two-stage HC MP process, it is difficult to address all the issues arising throughout this study. There stillexists a number of interesting topics and problems that can be investigated in the future work.The proposed control and estimation schemes have been tested on nonlinear process models that we de-veloped using a combination of empirical and mechanistic methods. However, the accuracy of the developedmodel is fundamental for ensuring high performance of the designed controllers. Therefore, an importantpart of the future work shall focus on validating and improving the accuracy of the developed model againstthe true two-stage HC MP process through a number of onsite trials.In this work, we can achieve the desirable pulp quality, which is measured at the downstream units of theMP process, by providing appropriate setpoints for the state variables to track, such as the production rate,the primary and secondary motor loads, and the primary and secondary consistencies. These setpoints arecalculated offline according to the requirement of the desirable pulp quality. One interesting future directionis to automate this procedure by designing two control loops in which the outer loop can measure pulpqualities, such as CSF, SC, and LFC, and compute the corresponding state setpoints which are passed to theinner loop. The inner loop employs the nonlinear econ MPC and steers the state variables to their setpointsreceived from the outer loop.Moreover, this project is a close collaborative work with our partner pulp mills. We will continue ourwork with the industry toward developing and applying the proposed control and estimation strategies inpulp mills. Based on the data we collected from the industry, data analysis will be conducted to furthermodify the nonlinear system models for the two-stage HC MP process. The internal model of the proposedm-econ MPC will be updated accordingly. 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Thesis/Dissertation
eng
Chemical and Biological Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Stochastic multi-objective economic model predictive control of two-stage high consistency mechanical pulping processes
Text