A tube feedback iterative learning control for batch processes

Proceedings, 10th IFAC International Symposium on Proceedings, 10th IFAC International Symposium on Advanced Control Chemical Processes Proceedings, 10th of IFAC International Symposium on at www.sciencedirect.com Proceedings, 10th IFAC International Symposium on Available online Advanced Control of Chemical Processes Proceedings, 10th of IFAC International Symposium on Shenyang, Liaoning, China, July 25-27, 2018 Advanced Chemical Processes Advanced Control Control of Chemical Processes Shenyang, Liaoning, China, July 25-27, 2018 Advanced of China, Chemical Processes Shenyang,Control Liaoning, China, July 25-27, 2018 2018 Shenyang, Liaoning, July 25-27, Shenyang, Liaoning, China, July 25-27, 2018

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IFAC PapersOnLine 51-18 (2018) 785–790

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feedback iterative learning control feedback feedback iterative iterative learning learning control control for batch processes feedback iterative learning control for batch processes for batch processes for batch processes ∗ ∗∗∗ ∗∗∗∗ Jingyi Lu ∗ Zhixing Cao ∗∗ ∗∗ Ridong Zhang ∗∗∗ Cuimei Bo ∗∗∗∗

Jingyi Lu ∗∗ Zhixing Cao ∗∗ Zhang ∗∗∗ Cuimei Bo ∗∗∗∗ ∗∗ Ridong ∗∗∗ Jingyi Lu Lu ∗ Zhixing Cao Ridong Zhang ∗∗∗ Cuimei Bo Bo ∗∗∗∗ Furong Gao ∗∗∗ Jingyi Zhixing Cao Zhang Cuimei ∗∗ Ridong ∗∗∗ Furong Gao ∗∗∗ Jingyi Lu Zhixing Cao Ridong Zhang Cuimei Bo ∗∗∗∗ ∗∗∗ Furong Gao Furong Gao ∗∗∗ Furong Gao ∗ ∗ Fok Ying Tung Graduate School, Hong Kong University of Science Fok Ying Tung Graduate School, Hong Kong University of Science ∗ ∗ Fok Ying Tung Graduate School, Kong and School, Hong HongChina Kong University University of of Science Science ∗ Fok Ying Tung Graduate and Technology, Technology, China ∗∗ Ying Tung Graduate Fok School, Hong Kong of University of Science andSciences, Technology, China University Edingurgh, UK ∗∗ School of Biological and Technology, China School of Biological Sciences, University of Edingurgh, UK ∗∗ ∗∗∗ ∗∗ andSciences, Technology, China School of of Biological Biological Sciences, University of Edingurgh, Edingurgh, UK Department of Chemical and Biological Engineering, Hong Kong ∗∗∗ ∗∗ School University of UK Department of Chemical and Biological Engineering, Hong Kong ∗∗∗ School of Biological Sciences, University of Edingurgh, UK ∗∗∗ Department of Chemical and Biological Engineering, Hong Kong University of Science and Technology, China, (e-mail: [email protected]) Department of Chemical and Biological Engineering, Hong Kong ∗∗∗ University of Science and Technology, China,Engineering, (e-mail: [email protected]) ∗∗∗∗ Department of Chemical and Biological Hong Kong University of Science and Technology, China, (e-mail: [email protected]) College of Automation and Electrical Engineering, Nanjing Tech ∗∗∗∗ University ofofScience and Technology, China, (e-mail: [email protected]) College Automation and Electrical Engineering, Nanjing Tech ∗∗∗∗ University of Science and Technology, China, (e-mail: [email protected]) ∗∗∗∗ College of Automation and Electrical Engineering, Nanjing Tech University, Nanjing 211816,China College of Automation and Electrical Engineering, Nanjing Tech ∗∗∗∗ University, Nanjing 211816,China College of Automation Electrical Engineering, Nanjing Tech University,and Nanjing 211816,China University, Nanjing 211816,China University, Nanjing 211816,China Abstract: Abstract: Optimization-based Optimization-based iterative iterative learning learning control control (OILC) (OILC) has has been been widely widely applied applied to to Abstract: Optimization-based iterative learning learning control (OILC) has been been widely applied to batch processes due to its fast convergence, good control performance and ability to handle Abstract: Optimization-based iterative control (OILC) has widely applied to batch processes due to its fast convergence, good control performance and ability to handle Abstract: Optimization-based iterative learning control (OILC) has convergence been widely applied to batch processes due to its fast convergence, good control performance and ability to handle constraints. However, how to guarantee constraint satisfaction and of tracking batch processes due to its fast convergence, good control performance and ability to handle constraints. However, how to guarantee constraint satisfaction and convergence oftotracking batch processes due to its fast convergence, good control performance and ability handle constraints. However, how to guarantee constraint satisfaction and convergence of tracking error in the presence of unknown system nonlinearity remains open in the framework of OILC. constraints. However,ofhow to guarantee constraint satisfaction and of oftracking error in the presence unknown system nonlinearity remains open in convergence the framework OILC. constraints. However, to guarantee constraint satisfaction and convergence of of error in the the presence presence ofhow unknown system nonlinearity remains open incommon the framework framework oftracking OILC. It is important to address this issue since unknown nonlinearity is in practice and error in of unknown system nonlinearity remains open in the OILC. It is important to address this issue since unknown nonlinearity is common in practice and error in the presence of unknown system nonlinearity remains openis inacommon the framework of OILC. It is important to address this issue since unknown nonlinearity in practice and detrimental to good control performance. In this paper, we propose tube feedback OILC to It is important to address this issue since unknown nonlinearity is common in practice and detrimental to good control performance. In this paper, we propose a tube feedback OILC to It is important to address this issue since unknown nonlinearity is common in practice and detrimental to good control performance. In this paper, we propose a tube feedback OILC to investigate the applicability of linear-model based control strategy on batch processes with an detrimental to good control performance. In this paper, we propose a tube feedback OILC to investigate the applicability of linear-modelInbased controlwe strategy ona batch processesOILC with an detrimental to good control performance. this paper, propose tube feedback to investigatenonlinear the applicability applicability of linear-model linear-model based control strategy on batch batch processes with an unknown term. First, a state feedback control law is designed to stabilize the system. investigate the of based control strategy on processes with an unknown nonlinear term. First, a state feedback control law is designed to stabilize the system. investigate the applicability of linear-model based control strategy on batch processes with an unknown nonlinear term. First,decomposed a state state feedback feedback control law is is designed designed to stabilize stabilize the system. system. The stabilized system is into two repeatable and unknown nonlinear term. First, a law to the The stabilized system is then then into control two subsystems: subsystems: repeatable and unrepeatable unrepeatable unknown nonlinear term. First,decomposed a state feedback control law is designed to stabilize the system. The stabilized system is then decomposed into two subsystems: repeatable and unrepeatable subsystems; Second, an invariant set of states corresponding to the unrepeatable subsystem The stabilized system is then decomposed into two subsystems: repeatable and unrepeatable subsystems; Second, anis invariant set of states corresponding to repeatable the unrepeatable subsystem The stabilized system then decomposed into two subsystems: and unrepeatable subsystems; Second, an invariant set of states corresponding to the unrepeatable subsystem is computed, based on which an OILC is further developed for the repeatable subsystem to subsystems; anwhich invariant set of isstates corresponding to the the repeatable unrepeatable subsystem is computed,Second, based on an OILC further developed for subsystem to subsystems; Second, anwhich invariant set of is corresponding to the the steers unrepeatable subsystem is computed, computed, based performance. on which an Meanwhile, OILC isstates further developed for the repeatable subsystem to improve the control the feedback controller the states within aa is based on an OILC further developed for repeatable subsystem improve the control performance. Meanwhile, the feedback controller steers the states withinto is computed, based on which an OILC is further developed for the repeatable subsystem to improve the control control performance. Meanwhile, the feedback feedback controller steers the the satisfaction states within within tube around the trajectory of OILC. In this way, convergence and constraint are improve the performance. Meanwhile, the controller steers states aa tube around the trajectory of OILC. In this way, convergence and constraint satisfaction are improve the control performance. Meanwhile, the feedback controller steers the states within a tube around the trajectory of OILC. In this way, convergence and constraint satisfaction are ensured simultaneously. Compared with the currently existing methods, the proposed method tube around the trajectory of OILC. In this way, convergence and constraint satisfaction are ensured simultaneously. Compared with the currently existing methods, the proposed method tube around the trajectory of OILC. In this way, convergence and constraint satisfaction are ensured simultaneously. Compared with the currently existing methods, the proposed method has the following advantages: (1) generality covering both stable and unstable systems; (2) low ensured simultaneously. Compared with thecovering currentlyboth existing the proposed has the following advantages: (1) generality stablemethods, and unstable systems; method (2) low ensured simultaneously. Compared with the currently existing the proposed method has the the following following advantages: (1) rigorous generality covering both stablemethods, and unstable systems;molding (2) low low computation complexity; and (3) stability. The simulation results on injection has advantages: (1) generality covering both stable and unstable systems; (2) computation complexity; and (3) rigorous stability. The simulation results on injection molding has the following advantages: (1) rigorous generality covering both stable and unstable systems;molding (2) low computation complexity; and that (3) rigorous stability. The simulation results on injection injection molding velocity control demonstrate the proposed method has superior performance. computation complexity; and (3) stability. The simulation results on velocity control demonstrate the proposed method has superior performance. computation complexity; and that (3) rigorous stability. The simulation results on injection molding velocity demonstrate that the method has performance. velocity control control demonstrate that the proposed proposed method has superior superior performance. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. velocity control demonstrate that the proposed method has superior performance. Keywords: Keywords: batch batch process, process, nonlinear, nonlinear, iterative iterative learning learning control, control, invariant invariant set, set, convergence. convergence. Keywords: Keywords: batch batch process, process, nonlinear, nonlinear, iterative iterative learning learning control, control, invariant invariant set, set, convergence. convergence. Keywords: batch process, nonlinear, iterative learning control, invariant set, convergence. 1. INTRODUCTION Model mismatch in OILC design is often often assumed assumed to to be be 1. INTRODUCTION Model mismatch in OILC design is 1. Model mismatch in OILC design is often assumed to be cycle-wise invariant and treated as repetitive disturbances, 1. INTRODUCTION INTRODUCTION Model mismatch in OILC design is often assumed to be cycle-wise invariant and treated as is repetitive disturbances, 1. INTRODUCTION Model mismatch in OILC design oftenthis assumed to be cycle-wise invariant and treated as repetitive disturbances, as shown in Foss et al. (1995). However, assumption cycle-wise invariant and treated as repetitive disturbances, shown in Foss et and al. (1995). However, thisdisturbances, assumption Batch process process plays plays an an important important role role in in manufacture manufacture and and as cycle-wise invariant treated as repetitive Batch as in Foss et (1995). However, assumption does not hold hold if the the model mismatch is this induced by linlinas shown shown in Foss et al. al. (1995). However, this assumption Batch process plays an important role in manufacture and does not if model mismatch is induced by chemical industry with high-value added due to its versaBatch process plays an important role in manufacture and as shown in Foss et al. (1995). However, this assumption chemical industry with high-value added due to its versadoes not hold if the model mismatch is induced by linearization. It is well known that model mismatch induced does not hold if the model mismatch is induced by Batch process plays an important role in manufacture and chemical industry with high-value added due to its versaearization. It is well known that model mismatch induced tility. Precise control of high-value key variables variables in the the process has aa does not hold if the model mismatch is induced by linchemical industry with added dueprocess to its versalintility. Precise control of key in has earization. It known model mismatch by linearization depends onthat system states and in in induced general earization. It is is well well known that model mismatch induced chemical industry with high-value added due to its versatility. Precise control of key variables in the process has linearization depends on system states and general significant impact on product quality. A growing attention tility. Precise control of key variables in growing the process has aa by earization. It is well known that model mismatch induced significant impact on product quality. A attention by depends on and not cycle-wise invariant. invariant. Researchers later developed developed algoby linearization linearization dependsResearchers on system system states states and in in general general tility. Precise of key variables inrecent the process has significant impact on product quality. growing attention cycle-wise later algois devoted to control batch process control inA years (Caoa not significant impact on process product quality.in A growing attention by linearization depends on system states and in general is devoted to batch control recent years (Cao not cycle-wise invariant. Researchers later developed algorithms which are robust against bounded non-repetitive not cycle-wise invariant. Researchers later developed algosignificant impact on product quality. A growing attention is to process control years which are robustResearchers against bounded non-repetitive et al. (2014),Cao (2014),Cao et al. al. (2016b),Wang etrecent al. (2017),Wang (2017),Wang is devoted devoted to batch batch process control in in et recent years (Cao (Cao rithms not cycle-wise invariant. later developed algoet al. et (2016b),Wang al. rithms which are robust against bounded non-repetitive disturbances, such as Liu and Wang (2012). These methrithms which are robust against bounded non-repetitive is devoted to batch process control in recent years (Cao et al. (2014),Cao et al. (2016b),Wang et al. (2017),Wang disturbances, such as Liu and Wang (2012). These meth(2018)). However, the inherent nonlinearity in batch et al. (2014),Cao et al. (2016b),Wang et al. (2017),Wang rithms which are robust against bounded non-repetitive et al. However, the inherent nonlinearity in batch disturbances, such Liu and (2012). These ods can be be extended extended to handle unknown nonlinearity by disturbances, such as as to Liuhandle and Wang Wang (2012). These methmethet al. (2018)). (2014),Cao et al. (2016b),Wang et al. (2017),Wang et (2018)). However, the nonlinearity in batch can unknown nonlinearity by processes introduces difficulty to control control strategy design: et al. al. (2018)). However, the inherent inherent nonlinearity indesign: batch ods disturbances, such mismatch as to Liuhandle andasWang (2012). These methprocesses introduces difficulty to strategy ods can be extended unknown nonlinearity by treating the model a combination of repetitive ods can be extended to handle unknown nonlinearity by et al. (2018)). However, the inherent nonlinearity in batch processes control design: the model mismatch as aunknown combination of repetitive controllers designed difficulty based on on to nonlinear model, such such as treating processes introduces introduces difficulty to control strategy strategy design: ods can be extended to handle nonlinearity by controllers designed based nonlinear model, as treating the model mismatch as aa combination of repetitive and non-repetitive disturbances and assuming the nontreating the model mismatch as combination of repetitive processes introduces difficulty to control strategy design: controllers designed based on nonlinear model, such as non-repetitive disturbances and assuming the nonnonlinear model predictive control, have intensive intensive on-line controllersmodel designed based control, on nonlinear model, such as and treating the model mismatch as a combination of repetitive nonlinear predictive have on-line and non-repetitive disturbances and assuming the nonrepetitive part has a fixed upper bound. To guarantee and non-repetitive assuming the noncontrollers designed based on nonlinear model, such as repetitive nonlinear predictive control, have on-line part has disturbances a fixed upperand bound. To guarantee computation (Nagy and Braatz (2003),Jia et al. al. (2013)); (2013)); nonlinear model model predictive control, have intensive intensive on-line and non-repetitive assuming the noncomputation (Nagy and Braatz (2003),Jia et repetitive part aaupper fixed upper bound. To guarantee the existence ofhas thedisturbances bound, cycle-wise differences repetitive partof has fixed bound, upperand bound. To differences guarantee nonlinear model predictive control, have intensive on-line computation (Nagy and Braatz (2003),Jia et al. (2013)); the existence the upper cycle-wise contrarily, controllers designed based on a linear model can computation (Nagy and Braatz (2003),Jia et al. (2013)); repetitive part has a fixed upper bound. To guarantee contrarily, controllers designed based on a linear model can the existence of the upper bound, cycle-wise differences on inputs andofstates states have to tobound, be regulated regulated intodifferences certain the inputs existence the upper cycle-wise computation (Nagy and Braatz (2003),Jia et al. (2013)); contrarily, controllers designed based on model can and have be into aa certain efficiently lower computation complexity, but the perforcontrarily, lower controllers designed based on aa linear linear model can on the existence of the upper bound, cycle-wise differences efficiently computation complexity, but the perforon inputs and states have to be regulated into aa certain region by hard constraints. This introduces a new question: on inputs and states have to be regulated into certain contrarily, controllers designed based on a linear model can efficiently lower computation complexity, but the perforby hard a into new question: mance is not not good enough due due to the the existent existent of perforsignifi- region efficiently lower computation complexity, but the on inputs and constraints. states have This toconsistently beintroduces regulated aafter certain mance is good enough to of signifiregion by hard constraints. This introduces aa new question: if the optimization problem feasible the region by hard constraints. This introduces new question: efficiently lower computation complexity, but the performance is not good enough due to the existent of signifiif the optimization problem consistently feasible after the cant model mismatch (Yang and Gao (2000)). To reduce mance is notmismatch good enough due toGao the (2000)). existent To of signifiregion by hard constraints. This introduces a new question: cant model (Yang and reduce if the optimization problem consistently feasible after the incorporation of those hard constraints? If not, control if the optimization problem consistently feasible after the mance is not good enough due to the existent of significant model mismatch Gao (2000)). reduce of those hardconsistently constraints?feasible If not,after control the model mismatch in the the and linear model-based design, cant model model mismatch mismatch (Yang (Yang and Gaomodel-based (2000)). To To design, reduce incorporation if the optimization problem the the in linear incorporation of those hard constraints? If not, control performance may degrade dramatically when infeasibility incorporation of those hard constraints? If not, control cant model mismatch (Yang and Gao (2000)). To reduce the in linear design, may degrade dramatically when infeasibility optimization-based iterative learning control is is preferred, preferred, the model model mismatch mismatch in the thelearning linear model-based model-based design, performance incorporation of those hard constraints? If not, control optimization-based iterative control performance may degrade dramatically when infeasibility occurs. The key issue considered in this paper is how performance degrade dramatically when infeasibility the model mismatch in the linear model-based design, occurs. optimization-based iterative learning control is The may key issue considered in this paper is how in the framework of which which model mismatch is compensated compensated optimization-based iterative learning control is preferred, preferred, performance degrade dramatically when infeasibility in the framework of model mismatch is occurs. The key issue considered paper is how to design anmay OILC which is able ablein tothis guarantee cycleoccurs. The key issue considered in this paper iscyclehow optimization-based iterative learning control is preferred, in the framework of which model mismatch is compensated to design an OILC which is to guarantee by prediction errors of previous cycles and control inputs in the framework of which model mismatch compensated occurs. Thean key issue considered into this paper despite iscyclehow by prediction errors of previous cycles andis control inputs to design OILC which is able guarantee wise convergence as well as consistent feasibility to design an OILC which is able to guarantee cyclein the framework of which model mismatch is compensated by of cycles and inputs convergence as well as is consistent feasibility despite are derived by byerrors optimizing a certain certain performance criteria. by prediction prediction errors of previous previous cycles and control controlcriteria. inputs wise to design an OILC which able to guarantee cycleare derived optimizing a performance wise convergence as well as consistent feasibility despite of unknown nonlinearity. It consistent is noted noted that that simultaneous wise convergence as well as feasibility despite by errors of previous cycles and control inputs of are derived optimizing aa certain performance criteria. unknown nonlinearity. It is simultaneous This idea has hasby been widely adopted (Lee and Lee Lee (2003),Shi are prediction derived bybeen optimizing certain performance criteria. wise convergence as well as consistent feasibility despite This idea widely adopted (Lee and (2003),Shi of unknown nonlinearity. It is noted that simultaneous guarantee of convergence and constraint satisfaction was of unknown nonlinearity. It is noted that simultaneous are derived by optimizing a certain performance criteria. This idea has been widely adopted (Lee and Lee (2003),Shi guarantee of convergence and constraint satisfaction was et al.idea (2006),Lu et widely al. (2015b),Lu (2015b),Lu et al. and (2018)). This has been adopted et (Lee Lee (2003),Shi guarantee of unknown nonlinearity. It isconstraint noted that simultaneous et al. (2006),Lu et al. al. (2018)). of convergence and satisfaction was guarantee of convergence and constraint satisfaction was This idea has been widely adopted (Lee and Lee (2003),Shi et et al. al. (2006),Lu (2006),Lu et et al. al. (2015b),Lu (2015b),Lu et et al. al. (2018)). (2018)). guarantee of convergence and constraint satisfaction was et al. (2006),Lu et al. (2015b),Lu et al. (2018)). 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Copyright © 2018 IFAC 779 Copyright 2018 responsibility IFAC 779Control. Peer review©under of International Federation of Automatic Copyright © 779 Copyright © 2018 2018 IFAC IFAC 779 10.1016/j.ifacol.2018.09.267 Copyright © 2018 IFAC 779

2018 IFAC ADCHEM 786 Shenyang, Liaoning, China, July 25-27, 2018 Jingyi Lu et al. / IFAC PapersOnLine 51-18 (2018) 785–790

studied in Lu et al. (2015a) and Lu et al. (2016). However, both of the works are for linear systems. In this paper, a tube feedback OILC scheme is proposed for reference tracking. The design borrows the idea of tube model predictive control, which is a well-known approach to handle bounded disturbances in MPC design (Langson et al. (2004)). First, model mismatch induced by linearization is decomposed into repeatable and unrepeatable parts, following which, the original system is decomposed into repeatable and unrepeatable subsystem accordingly. Similar to the tube MPC, a state feedback controller is then designed such that the states corresponding to the unrepeatable subsystem are regulated into an invariant tube. Second, an OILC which actually determines the center of the tube is further designed to reject the repeatable mismatch and enhance tracking performance from cycle to cycle. In this way, cycle-wise convergence and constraint satisfaction are ensured simultaneously. Albeit the advantage on explicit guarantee of convergence and feasibility, the proposed method has less restrictions than other existing OILC algorithms. The only requisite is that the system should be “stabilizable’, while many existing OILC algorithms have strict requirements of the controlled system. For example, a common type of OILC is designed based on an minimization of the predicted tracking error over the entire cycle, as shown in Lee et al. (2000) and Chin et al. (2004). Such an optimization is based on a lifted system model, which is only available when the system is stable. If the system subjects to unknown disturbances and constraints, this type of method can not be directly extended to unstable systems by stabilizing the system first. There are also methods (Liu and Wang (2012)) which are devised by solving linear matrix inequalities (LMIs) induced by a 2D Lyapunov function, which are limited by the feasibility of the LMIs. The rest of the paper is arranged as follows: Section 2 gives the formulation of the problem; Section 3 details the controller design; Section 4 provides stability analysis; Section 5 presents simulation results, and Section 6 draws the conclusions. 2. PROBLEM FORMULATION A nonlinear batch process can be represented by a linear model plus a nonlinear term (Gao et al. (2014)) as x(t + 1, k) =Ax(t, k) + Bu(t, k) + g(x(t, k), u(t, k)), y(t, k) =Cx(t, k). Here t ∈ [0, tn − 1] is the time index. tn is the length of a cycle. k ∈ [1, ∞) is the cycle index. x ∈ Rns is the system state. u ∈ Rni is the input. y ∈ Rno is the output. States and outputs are assume to be measurable or observable (Cao et al. (2016a), Cao et al. (2016c)). System dynamic matrices A, B and C are arranged in proper dimensions. The pair (A, B) is stabilizable. In addition, it is assumed that the initial states in each cycle are the same, namely x(0, k) = x(0, k − 1) = · · · = x(0, 1) = x0 . The nonlinear term g satisfies Lipschiz condition as g(u1 ) − g(u2 )∞ ≤ Lu1 − u2 ∞ , (1) with L a global Lipschiz constant. For simplicity of notations, it is assumed that g is solely a function of input u. Then, the system model is simplified as 780

x(t + 1, k) =Ax(t, k) + Bu(t, k) + g(u(t, k)), (2) y(t, k) =Cx(t, k). For more general cases, the computation will become more complex but the underlying idea remains the same. Define a polytopic set U in cycle k as U(k) = {u : u(t, k) − u(t, k − 1)∞ ≤ δ(k)}. (3) An input constraint is posed as u(t, k) ∈ U(k), (4) such that the cycle-wise variation of the nonlinear term g is bounded as g(u(t, k)) − g(u(t, k − 1))∞ ≤ Lδ(k) (5) according to (1). Define de (t, k) = g(u(t, k − 1)) = x(t + 1, k − 1) − Ax(t, k − 1) − Bu(t, k − 1). (6) and dδ (t, k) = g(u(t, k)) − g(u(t, k − 1)). It is noted that de and dδ split the nonlinear term into cycle-wise repeatable and unrepeatable parts. Moreover, in view of (4) and (5), dδ is bounded in the set F(k) as dδ (t, k) ∈ F(k) (7) with F(k) defined as (8) F(k) = {f ∈ Rns : −Lδ(k) ≤ fi ≤ Lδ(k)}. Here fi is the i-th coordinate of the vector f . Denote the tracking reference as yr (t) with t ∈ I[0,tn −1] . The control objective is to steer the output y(t, k) to a given reference yr (t) despite of the unknown terms de (t, k) and dδ (t, k), meanwhile, maintain the input confined to U(k). 3. CONTROLLER DESIGN In the section, details of the controller design are given. 3.1 System decomposition To guarantee time-wise stability, a state feedback gain K is selected to stabilize the system, namely keep the spectral radius ρ(A + BK) < 1, following which the structure of the controller is fixed as (9) u(t, k) = Kx(t, k) + u0 (t, k). The second term u0 (t, k) is to be induced by ILC for reference tracking and repeatable disturbance rejection. Define Ak = A + BK. The repeatable subsystem is constructed with respect to de (t, k) as x ¯(t + 1, k) =Ak x ¯(t, k) + Bu0 (t, k) + de (t, k). (10) By taking difference between (2) and (10), the unrepeatable subsystem is constructed as x ˆ(t + 1, k) =Ak x ˆ(t, k) + dδ (t, k), (11) with x ˆ(t, k) = x(t, k) − x ¯(t, k). Set x ¯(0, k) = x(0, k) = x0 , x ˆ(0, k) = 0. Accordingly, the inputs can also be decomposed into two parts. Define u ¯(t, k) = K x ¯(t, k) + u0 (t, k) as the input corresponding to the repeatable subsystem and u ˆ(t, k) = K x ˆ(t, k) (12) as the input corresponding to the non-repeatable subsystem. Furthermore, the tracking error satisfies

2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018 Jingyi Lu et al. / IFAC PapersOnLine 51-18 (2018) 785–790

e(t, k) =yr (t) − y(t, k) = yr (t) − C x ¯(t, k) − C x ˆ(t, k). (13) Thus, define Repeatable: e¯(t, k) = yr (t) − C x ¯(t, k) (14) and Unrepeatable: eˆ(t, k) = −C x ˆ(t, k). (15) To this end, the original system has been decomposed into two subsystems. 3.2 Preliminaries of invariant sets Note from (7) that dδ (t, k) ∈ F(k) for any t and k. Based on (11), according to Theorem 4.1 in Kolmanovsky and Gilbert (1998), it can be concluded that Lemma 1. Given that the spectral radius of A + BK less than 1, dδ (t, k) contained in the polyhedral set F(k) and x ˆ(0, k) stays at the origin, there exists a polyhedral robust positively invariant set Ωx (k) on x ˆ(t, k) in cycle k such that (i) for any x ˆ(t, k) ∈ Ωx (k), x ˆ(t + 1, k) satisfying (11) stays in Ωx (k); (ii) the set Ωx (k) contains the origin 0. In view of the item (ii) in Lemma 1, we have x ˆ(0, k) ∈ Ωx (k). Moreover, by recursively applying (i), it can be proved that x ˆ(t, k) ∈ Ωx (k) for any t. Remark 1. Generally, the invariant sets in Lemma 1 are not unique. In order to reduce conservativeness, the minimal robust positively invariant (mRPI) set is adopted. Methods on computing mRPIs are provided by Kolmanovsky and Gilbert (1998). Briefly speaking, a mRPI is computed based on sequential Minkowski summation: ∞  Ωx (k) = Aik BF(k). (16) i=0

Toolboxes shown in Herceg et al. (2013) can be used to compute the set.

According to (12), the set of u ˆ(t, k) can be characterized by a linear mapping as ˆ U(k) = {ˆ u = Kx ˆ:x ˆ ∈ Ωx (k)}. (17)

For any t and k it can be easily proved that u ˆ(t, k) is ˆ bounded in the set U(k) since Ωx (k) is a compact set and the mapping in (12) is continuous. Similarly, according to ˆ (15), eˆ(t, k) stays in E(k) which is a polyhedral set defined as ˆ (18) E(k) = {ˆ e = Cx ˆ:x ˆ ∈ Ωx (k)}. Since u ¯(t, k) and u ˆ(t, k) are governed by u(t, k) = u ¯(t, k) + u ˆ(t, k), to guarantee input constraint fulfillment, u ¯(t, k) ¯ should be steered to a set U(k) according to Lemma 2. ˆ Lemma 2. Given a u ˆ(t, k) ∈ U(k), if u ¯(t, k) satisfies ¯ ¯ ˆ u ¯(t, k) ∈ U(k) with U(k) = U(k) ∼ U(k), then it can be guaranteed that u(t, k) = u ˆ(t, k) + u ¯(t, k) ∈ U(k). Here ’∼’ denotes Pontryagin difference. This lemma can be directly proved by Theorem 2.1 in Kolmanovsky and Gilbert (1998). In addition, as stated in Section 3.1.2 in 781

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Blanchini and Miani (2007), C-set and 0-symmetric are defined as: Definition 1. (C-set): A C-set is a convex and compact subset of Rn including the origin as an interior point. Definition 2. (0-symmetric): A C-set S is 0-symmetric if x ∈ S ⇒ −x ∈ S. According to Definition 1 and 2, it can be checked that the set F(k) is a 0-symmetric C-set. Moreover, it can also be proved that ˆ ˆ Proposition 1. The set Ωx (k), U(k) and E(k) are all 0symmetric C-sets. 3.3 Design of OILC Section 3.2 tells that the states, inputs and tracking error corresponding to the unrepeatable subsystem are bounded ˆ and E ˆ respectively, based on which an in the set Ωx , U OILC can be applied to the repeatable subsystem and devised by a quadratic optimization. Problem 1.  min ¯ e(k)22 +  Rw ∆k u0 (k)22 δ(k),∆k u0 (t,k),¯ e(k)

s.t.

x ¯(t + 1, k) = Ak x ¯(t, k) + Bu0 (t, k) + de (t, k), t = 0, 1, . . . , tn , (19) ¯(t, k) (20) e¯(t, k) = yr (t) − C x ¯ Kx ¯(t, k) + u0 (t, k) − u(t, k − 1) ∈ U(δ(k)), (21) √ ¯ e(k)2 + tn ˆ e(k)∞ ≤ ¯ e(k − 1)2 √ + tn ˆ e(k − 1)∞ , (22) ¯(k) = [¯ e e(1, k)T e¯(2, k)T . . . e¯(tn , k)T ]T ,

u0 (k) = [u0 (0, k)T , . . . u0 (tn − 1, k)T ]T , ∆k u0 (k) = u0 (k) − u0 (k − 1). The first term in the objective function is to minimize the predicted tracking error corresponding to the repeatable subsystem over the entire cycle. The matrix Rw ∈ Rni ×ni is positive definite working as a penalty weight on the input term to regulate its variation. Eq. (19) and (20) give the system equations. The constraint in (21) is used to ensure input constraint satisfaction. The inequality (22) √ guarantees a monotonic decrease on ¯ e(k)2 + tn ˆ e(k)∞ which is an upper bound of the tracking error in cycle k. The variable δ(k) is used to determine the range of variation on ∆k u(t, k) such that the convergence condition in (22) can be satisfied. 3.4 Analysis of computation complexity In this section, we analyze the computation complexity of Problem 1. Proposition 2. Problem 1 is a quadratic constrained quadratic programming. ˆ Proof : According to Proposition 1, the set U(k), U(k) and ˆ E(k) are polyhedral C-sets, they can be denoted by linear inequalities as

2018 IFAC ADCHEM 788 Shenyang, Liaoning, China, July 25-27, 2018Jingyi Lu et al. / IFAC PapersOnLine 51-18 (2018) 785–790

y(k − 1) x(k − 1)

yr

− e(k − 1)

Memory

y(k) x(k)

u(k)

+ u0 (k) u(t, k)

OILC

Process x(t, k)

K

Fig. 1. Illustration of the control scheme. ˆ U(k) = {u : M0 u ≤ δ(k)N0 }, ˆ U(k) ∼ U(k) = {u : M1 u ≤ δ(k)N1 },

Proof : Assume Problem 1 is feasible in cycle k − 1. Then, in cycle k, taking δ(k) = 0, it can be derived ¯(t, k) = x ¯(t, k − 1) and that u0 (t, k) = u0 (t, k − 1), x e(t + 1, k) = e(t + 1, k − 1) for any t ∈ [0, tn − 1], which provide a group of feasible solution to the constraint in (21). Moreover, in view of (13), (14) and (15), the tracking error satisfies y(t, k) ¯(k − 1) + e ˆ(k − 1). e(k) = e(k − 1) = e According to the triangle inequality, it is further proved that e(k − 1)2 ≤¯ e(k − 1)2 + ˆ e(k − 1)2 (23) √ ≤¯ e(k − 1)2 + tn ˆ e(k − 1)∞ , which ensures that the inequality (22) holds. By explicitly giving a group of feasible solution to Problem 1 in cycle k, its feasibility is proved.

ˆ E(k) = {e : M2 e ≤ δ(k)N2 }, Thus, (21) is equivalent to M1 (K x ¯(t, k)+u0 (t, k)−u(t, k − 1)) ≤ δ(k)N2 . It is linear on the variable δ(k), x ¯(t, k) and ˆ u0 (t, k). Given eˆ(k) ∈ E(k), we have ˆ e(k)∞ =

sup

I j e.

ˆ j∈[1,no ], e∈E(k)

It can be further checked that ˆ e(k)∞ = δ(k)

sup

I j e.

j∈[1,no ], e∈{e:M2 e≤N2 }

Note that the term c1 = supj∈[1,no ], e∈{e:M2 e≤N2 } I j e is fixed after K is determined and therefore can be √ considered e(k − as a constant. Similarly, the term ¯ e(k − 1)2 + tn ˆ 1)∞ is also a constant since it is fixed when cycle k − 1 is finished. Denote it as c2 . The inequality in (22) is equivalent to ep (k)2 + δ(k)c1 ≤ c2 . which is a quadratic constraint. Since the objective function of Problem 1 is a quadratic function and the constraints are either linear or quadratic functions, Problem 1 is a quadratically constrained quadratic programming, which is convex and therefore can be efficiently solved by methods such as interior point method (Mehrotra (1992)). The overall control scheme is shown in Fig. 1. The state feedback control (dotted block) is executed at each sample time while the outer loop is conducted once a cycle. Thus, it is noted that the method enjoys a low computation complexity. 4. STABILITY ANALYSIS Similar to many MPC algorithms, feasibility can imply stability in our problem. Thus, we study the consistent feasibility of Problem 1 first. Theorem 1. (Feasibility to constraints): If the constraint in (21) is satisfied, then satisfaction of (4) is guaranteed. Proof : The proof is similar as Lemma 2. Since u ˆ(t, k) ∈ ˆ ˆ U(k) and K x ¯(t, k) + u0 (t, k) − u(t, k − 1) ∈ U(k) ∼ U(k), we have u(t, k)− u(t, k −1) = u ˆ(t, k)+ K x ¯(t, k) +u0 (t, k) − ˆ 1 . Thus, (4) is ensured. u(t, k − 1) ∈ U1 ∼ U Theorem 2. (Recursive feasibility): If Problem 1 is feasible in cycle k − 1, then it is also feasible in cycle k. 782

Next, we can prove the boundedness of tracking error based on its feasibility. Theorem 3. (Bounded tracking error): By applying the control law in (9), the tracking error e(k)2 is bounded in each cycle. Proof : This property is guaranteed √ by the inequality (22). e(k)∞ is an upper According to (23), ¯ e(k)2 + tn ˆ bound of e(k)2 . The constraint in (23) guarantees that the upper bound of each cycle is non-increasing. Thus, e(k)2 for each cycle is bounded. Remark 2. According to Theorem 3, the controller guarantees the decrease of an upper bound of the tracking error, instead of the tracking error itself. If a monotonic decrease on tracking error is desired, (23) can be replaced √ e(k)∞ ≤ e(k−1)2 . However, this conby ¯ e(k)2 + tn ˆ straint brings conservativeness, which means the variable δ(k) may quickly go to 0. In this case, control performance can hardly get improved even though monotonic decrease of tracking error is guaranteed. Problem 1 uses (23) to guarantee the robust stability and optimize the tracking performance based on the nominal system. Shown by Xu et al. (2012), this approach can help to avoid conservativeness. 5. SIMULATIONS In this section, control of injection velocity in an injection molding process, which is an important polymer processing technique, is taken as an example. In velocity control, ram velocity is controlled by hydraulic pressure. The mechanism model is given in Cao et al. (2015) as P˙h =βh /Vh (qh − Ah vz ),

v˙ z =1/M (Ph Ah − Pn An − fv ), P˙ n =βp /Vn (An vz − qp ). Here Ph denotes hydraulic pressure. Pn denotes nozzle pressure. Vh and Vn are volumes in cylinder and nozzle respectively. Ah and An are cross-section areas of hydraulic system and nozzle. fv is the friction force. qh is hydraulic flow rate and qp is polymer melt flow rate. βh and βp are hydraulic oil and polymer melt bulk modulus. M is ram mass. In Cao et al. (2015), it was shown that βh /Vh and βp /Vn can be treated as constants. Under the air shot operation, Pn is also a constant. Thus, the map between qh and vz is linear by taking Pn An and fv as exogenous

2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018Jingyi Lu et al. / IFAC PapersOnLine 51-18 (2018) 785–790

disturbances. This has been verified by experiments (Yang (2004)) and the following state space model is used in this work.     1.6 −0.5916 1 x(t + 1, k) = x(t, k) + u(t, k) + n(t, k), 1 0 0

40 30 Output

y(t, k) = [ 1.69 1.419 ] x(t, k). In this simulation, the nonlinear term n(k) is set as   T T n(t, k) = 0.2 u(t, k)2 + 1 0.2|u(t, k)|T . The reference to be tracked  is set as 15, t ∈ [1, 20] yr (t) = 30, t ∈ [21, 50].

20 Cycle 1 Cycle 5 Cycle 10

10

For comparison, the method in Shi et al. (2006) is implemented. This robust control strategy was designed by solving linear matrix inequalities induced by a twodimensional Lyapunov function. Parameters are taken as  = 0.1, γ = 2.5, ρk = 0.1 and ρt = 0.1. To compare the cycle-wise convergent rate, the method in Shi et al. (2006) is also adopted to control the first cycle when the proposed method is simulated such that the outputs of the two methods are the same in cycle 1. Fig. 2 and Fig. 3 gives the outputs in cycle 1, 5 and 10 of the two methods, showing that the proposed method has a superior performance with less oscillations. This is further verified by Fig. 4 in which the mean square tracking errors (MSE) of the two methods are plotted. The MSE of the proposed method is significantly reduced by the proposed method.

40

0 0

2

20

Time

30

40

50

Shi’s method proposed method

1.5

1

0.5

0 0

30

10

Fig. 3. The method in Shi et al. (2006): Outputs of cycle 1, 5 and 10

Mean Squre Error

The eigenvalue of linear part is 1.02 and 0.58, which shows the system unstable. A state feedback gain K = [ −1.6 0.5916 ] is selected to stabilize the system, based on ˆ and U − U ˆ can be computed. Then, take which, the set U Qw = diag(1, 1) and Rw = diag(0.01, 0.01). System inputs for each cycle can be derived by Problem 1.

Output

789

10

20 30 Cycle

40

50

Fig. 4. Comparison on mean square error of the proposed method and Shi’s method in Shi et al. (2006)

20 Cycle 1 Cycle 5 Cycle 10

10 0 0

of a feedforward term and a feedback term. The feedforward term, derived by optimal iterative learning control, improves the tracking performance. The feedback term, induced by a state feedback controller, stabilizes the entire system and regulates the states within a tube around the trajectory determined by the OILC. In this way, constraint fulfillment and cycle-wise tracking error decrease are ensured simultaneously. In sum, the proposed method has low online computation complexity and wide applications covering both stable and unstable systems.

10

20

Time

30

40

50

Fig. 2. Proposed method: Outputs of cycle 1, 5 and 10. ACKNOWLEDGEMENTS 6. CONCLUSIONS In this paper, a tube feedback iterative learning control strategy is designed for batch processes with unknown nonlinearity. Intrinsically, the control input is composed 783

The authors gratefully acknowledge the financial support from Hong Kong Research Grants Council(No.16233316) and the National Natural Science Foundation of China project (No. 61433005).

2018 IFAC ADCHEM 790 Shenyang, Liaoning, China, July 25-27, 2018 Jingyi Lu et al. / IFAC PapersOnLine 51-18 (2018) 785–790

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