Iterative Learning Stabilization and Fault-Tolerant Control for Batch Processes 9789811357909, 9811357900

Table of contents :
Introduction --
Iterative learning control of linear batch processes --
Iterative learning control of nonlinear batch processes --
Iterative learning optimal guaranteed cost control of batch processes --
Iterative learning control of multi-phase batch processes --
Delay-dependent iterative learning control of multi-phase batch processes --
Iterative learning fault-tolerant control of linear batch processes --
Iterative learning fault-tolerant control of nonlinear batch processes --
Iterative learning fault-tolerant control of multi-phase batch processes --
Further ideas on constrained infinite horizon fault-tolerant control of batch processes.

Citation preview

Limin Wang · Ridong Zhang · Furong Gao

Iterative Learning Stabilization and Fault-Tolerant Control for Batch Processes

Iterative Learning Stabilization and Fault-Tolerant Control for Batch Processes

Limin Wang Ridong Zhang Furong Gao •

•

Iterative Learning Stabilization and Fault-Tolerant Control for Batch Processes

123

Limin Wang College of Mathematics and Statistics Hainan Normal University Haikou, Hainan, China

Ridong Zhang Hangzhou Dianzi University Hangzhou, Zhejiang, China

Furong Gao Department of Chemical and Biomolecular Engineering Hong Kong University of Science and Technology Hong Kong, China

ISBN 978-981-13-5789-3 ISBN 978-981-13-5790-9 https://doi.org/10.1007/978-981-13-5790-9

(eBook)

Library of Congress Control Number: 2018965452 © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Batch Process Control . . . . . . . . . . . . . . 1.2 Current Status of Batch Process Control . 1.3 The Main Content of This Book . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Iterative Learning Control of Linear Batch Processes . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 2D Roesser Model-Based Iterative Learning Control for Batch Processes with Uncertainties and Interval Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Equivalent 2D System Representation . . . . . . . . . . . 2.2.3 Robust Stability and Control of the 2D System . . . . 2.3 2D-FM Model-Based Iterative Learning Control for Batch Processes with Uncertainties and Interval Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Equivalent 2D System Representation . . . . . . . . . . . 2.3.2 Robust 2D Controller Design and System Structure . 2.3.3 Design Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Delay-Range-Dependent Robust 2D Output Feedback Iterative Learning Control for Batch Processes with State Delay and Uncertainties Based on 2D-FM Model . . . . . . . . . . . . . . 2.4.1 Equivalent 2D System Representation . . . . . . . . . . . 2.4.2 Robust Stability and Control of a 2D-FM System . . . 2.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Design Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Design Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Iterative Learning Control of Multi-phase Batch Processes . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Iterative Learning Control for Multi-phase Batch Processes . . 4.2.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Hybrid 2D ILC Design (Without Disturbance) . . . . . 4.2.3 Robust 2D ILC Design (With Disturbance) . . . . . . . 4.2.4 Performance Optimization . . . . . . . . . . . . . . . . . . . . 4.3 Iterative Learning Control for Multi-phase Batch Processes with Time Delay and Disturbances . . . . . . . . . . . . . . . . . . . . 4.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Robust Hybrid 2D Iterative Learning Control Design Based on Time Delay in a Range . . . . . . . . . . . . . . 4.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Design Case 1 (Without Disturbance) . . . . . . . . . . . 4.4.2 Robust Analysis (With Disturbances) . . . . . . . . . . . . 4.4.3 Design Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Iterative Learning Control of Nonlinear Batch Processes . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 2D T-S Fuzzy Model-Based Iterative Learning Control for Nonlinear Batch Processes . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Establishment of the 2D T-S Fuzzy Model . . . . . . . 3.2.2 Robust Fuzzy Iterative Learning Controller Design . 3.3 2D T-S Fuzzy Model-Based Iterative Learning Control for Batch Processes with Interval Time-Varying Delays . . . 3.3.1 Establishment of the 2D T-S Fuzzy Model . . . . . . . 3.3.2 Fuzzy Iterative Learning Controller Design . . . . . . 3.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Design Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Design Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Iterative Learning Optimal Guaranteed Cost Control of Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Guaranteed Cost Control for Batch Processes with Time-Varying Delay . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Problem Description and 2D System Representation . 5.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Output Feedback-Based Iterative Learning Optimal Guaranteed Cost Control of Batch Processes with State Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Problem Description and 2D System Representation . Iterative Learning Guaranteed Cost Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Design Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 A Suboptimal Guaranteed Cost Control for Multi-phase Batch Processes with Time-Varying Delay . . . . . . . . . . . . . . . . . . . 5.4.1 Problem Description and 2D System Representation . 5.4.2 Delay-Range-Dependent Robust Hybrid 2D Iterative Learning Control Design . . . . . . . . . . . . . . . . . . . . . 5.5 2D Fuzzy Guaranteed Cost Control Strategy for Nonlinear Batch Processes with Time-Varying Delay . . . . . . . . . . . . . . 5.5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 The Design of Fuzzy Iterative Learning Controller with Optimal Control Performance . . . . . . . . . . . . . . 5.6 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Design Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Design Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Design Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Design Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Iterative Learning Predictive Control for Batch Processes . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 2D-FM Model-Based Robust Iterative Learning Predictive Control for Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Problem Description and 2D-FM System Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Stability Analysis and Controller Design of a 2D System . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 2D Fuzzy Constrained Predictive Control of Nonlinear Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Question Description and Modeling . . . . . . . . . . . . . 6.3.2 Design of 2D T-S Fuzzy Iterative Learning Predictive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 2D-FM Model-Based Robust Iterative Learning Predictive Control for Multi-Phase Batch Processes . . . . . . . . . . . . . . . 6.4.1 Equivalent 2D System Representation . . . . . . . . . . . 6.4.2 Stability Analysis and Controller Design of a 2D System . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.5

Case Study . . . 6.5.1 Design 6.5.2 Design 6.6 Conclusion . . . References . . . . . . . .

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Iterative Learning Fault-Tolerant Control of Linear Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Robust Delay-Dependent Iterative Learning Fault-Tolerant Control for Batch Processes with State Delay and Actuator Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Robust 2D FILRC Design . . . . . . . . . . . . . . . . . . . . 7.3 Delay-Range-Dependent Method for Iterative Learning Fault-Tolerant Guaranteed Cost Control for Batch Processes . 7.3.1 Equivalent 2D Representation . . . . . . . . . . . . . . . . . 7.3.2 ILRG Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Design Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Design of Fault-Tolerant Guaranteed Performance Controller for Batch Process Compound Iterative Learning . . . . . . . . . . 7.4.1 Equivalent Two-Dimensional Description . . . . . . . . . 7.4.2 Iterative Learning Reliable Guaranteed Performance Control Law Design . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Design Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Design Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iterative Learning Fault-Tolerant Control of Nonlinear Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Iterative Learning Fault-Tolerant Control for Batch Processes Based on the T-S Fuzzy Model . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Equivalent 2D Representation . . . . . . . . . . . . . . . . . 8.3 Design of Fuzzy Iterative Learning Fault-Tolerant Control for Batch Processes with Time-Varying Delays . . . . . . . . . . . 8.3.1 Equivalent 2D Description . . . . . . . . . . . . . . . . . . . 8.4 Fuzzy Iterative Learning Control-Based Design of Fault-Tolerant Guaranteed Cost Controller for Nonlinear Batch Processes with Time-Varying Delays . . . 8.4.1 The Design of Optimal Cost-Guaranteed Controller Based on Fuzzy Iterative Learning Control . . . . . . . . 8.4.2 Design Algorithms . . . . . . . . . . . . . . . . . . . . . . . . .

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Case Study . . . 8.5.1 Design 8.5.2 Design 8.5.3 Design 8.6 Conclusion . . . References . . . . . . . .

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Iterative Learning Fault-Tolerant Control of Multi-phase Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Robust Iterative Learning Fault-Tolerant Control for Multi-phase Batch Processes with Uncertainties . . . . . . . . 9.2.1 Traditional Reliable Control (TRC) . . . . . . . . . . . . . 9.2.2 Iterative Learning Reliable Control (ILRC) . . . . . . . . 9.2.3 Algorithm Design . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 A Hybrid 2D Fault-Tolerant Controller Design for Multi-phase Batch Processes with Time Delay . . . . . . . . . . . . . . . . . . . . 9.3.1 Equivalent 2D Expression . . . . . . . . . . . . . . . . . . . . 9.3.2 Design of Delay-Range-Dependent Robust Hybrid 2D Iterative Learning Fault-Tolerant Control Law . . . . . 9.4 Delay-Range-Dependent-Based Hybrid Iterative Learning Fault-Tolerant Guaranteed Cost Control for Multi-phase Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Equivalent 2D Expression . . . . . . . . . . . . . . . . . . . . 9.4.2 Robust H1 Guaranteed Cost Performance Analysis . 9.4.3 Performance Optimization . . . . . . . . . . . . . . . . . . . . 9.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Design Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Design Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Design Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Further Ideas on Constrained Infinite Horizon Fault-Tolerant Control of Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Robust Constraint Iterative Learning Predictive Fault-Tolerant Control of Uncertain Batch Processes . . . . . . . . . . . . . . . . . . 10.2.1 Equivalent 2D Model . . . . . . . . . . . . . . . . . . . . . . . 10.3 2D Fuzzy-Constrained Fault-Tolerant Predictive Control of Nonlinear Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Design of Optimal Controller . . . . . . . . . . . . . . . . .

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10.4 2D-FM Model-Based Robust Iterative Learning Fault-Tolerant Predictive Control for Multi-phase Batch Processes . . . . . . . . 10.4.1 Equivalent 2D Model . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Stability Analysis and Controller Design of a 2D System . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Design Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Design Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

1.1 Batch Process Control In recent years, batch processes, which are one of the important production methods in industrial processes, are widely used in fine chemicals, medicine production, biological products, and modern agriculture. In order to meet the market requirements of multiple varieties, multiple specifications, and high quality, batch processes have received more and more attention. Research on optimization and advanced control of batch process has also appeared as a new research area [1–32]. In the actual production process, disturbances and uncertainty are often widespread and unavoidable due to the influence of environment and inaccurate modeling. Their existence will not only lead to unstable operation of the system, but also affect product quality [33–56]. Compared to disturbances and uncertainties, actuator failures and time delay may have a greater impact on system stability and control performance [57–78]. Actuator failures will not only reduce the control accuracy of the system, but also damage the control performance of the system. In severe cases, it will cause damage to the production equipment and even threaten the personal safety. Similarly, this is extremely unfavorable for safe production of the products. The existence of hysteresis will cause the delay of the response speed of the system, as well as the deterioration of control performance, which will eventually lead to the reduction of production efficiency and is contrary to the current “high-efficiency and energy-saving” industrial development requirements. Therefore, how to improve the reliability of the process systems in the event of actuator failures and time delay and improve the control performance is an urgent problem to be solved.

© Springer Nature Singapore Pte Ltd. 2020 L. Wang et al., Iterative Learning Stabilization and Fault-Tolerant Control for Batch Processes, https://doi.org/10.1007/978-981-13-5790-9_1

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1 Introduction

1.2 Current Status of Batch Process Control A batch process is an industrial process in which a limited amount of raw material is processed into a product in a limited time, using one or more equipment, in accordance with specified production requirements and fixed processing steps. Due to different production environments and methods, batch processes have their unique characteristics that are different from continuous processes: (1) repeatability; (2) two dimensionality; (3) multi-phase; (4) limited runtime; (5) low cost. Similarly, the following difficulty exists in the study of batch process control: (1) nonlinearity; (2) significant time-varying characteristics; (3) information delay. In view of the characteristics of the above batch processes and the control difficulty, some research results have appeared at this phase, which are summarized as follows. From the system property point of view, batch processes can be divided into linear systems and nonlinear systems; from the processing procedure, batch processes can be divided into single-phase processes and multi-phase processes. Compared with single-phase batch processes, multi-phase batch processes are more complicated in the establishment of the models. Since each phase of the multi-phase batch process is closely related to one another, the control and operation at the current phase will have a direct impact on the next phase, which adds difficulty to the study of the multi-phase batch process. In addition, the control objectives of different phases in the batch process are different, which makes the design of the control law more complicated. However, whether it is a single-phase process or a multi-phase process, the system model description is inseparable from the following two situations: (1) Linear system models 

x(t + 1, k)  Ax(t, k) + Bu(t, k) y(t, k)  C x(t, k)

(1.1)

Equation (1.1) is a relatively common batch process system model, where x(t, k) ∈ R n , u(t, k) ∈ R m , and y(t, k) ∈ R l represent the state, input, and output of the system at the t time of the k batch, and A, B, C are matrices with suitable dimensions. (2) Nonlinear system models 

x(t + 1, k)  f [x(t, k), u(t, k)] y(t, k)  g[x(t, k)]

(1.2)

where f (x(t, k), u(t, k)) represents a nonlinear function related to the system state and the control inputs; g(x(t, k)) represents a nonlinear function related to the state of the system.

1.2 Current Status of Batch Process Control

3

In the initial phase of batch process control, batch processes are generally regarded as continuous processes for control. The control strategies used are mainly PID control [79–90] and adaptive control [91–106]. Since it is difficult for PID control to achieve the control requirements for systems with nonlinearity and time delays, new methods were appearing gradually. The iterative learning control method is one of these [107–133], which makes full use of the repeating characteristics of the batch process, and is especially suitable for the control of systems that is nonlinear, difficult to be modeled and with high-precision trajectory tracking requirement. However, the pure iterative learning control has extremely poor anti-interference performance and is not suitable for non-minimum phase systems [134, 135], as well as time-varying uncertain systems [136, 137]. In order to solve this problem, the papers [138–152] combined feedback control with iterative learning control and formed a composite iterative learning control strategy, which greatly reduced the influence of disturbance and improved the stability of the closed-loop system. The Roesser model [153] and the Fornasini-Marchesini (FM) model [154] are two 2D state space models commonly used in iterative learning control. The specific forms are: (1) 2D Roesser model ⎧       A11 A12 xh (i, j) B1 xh (i + 1, j) ⎪ ⎪ u(i, j)  + ⎪ ⎨ xv (i, j + 1) A21 A22 xv (i, j) B2 , i, j > 0   ⎪

xh (i, j) ⎪ ⎪ y(i, j)  C1 C2 ⎩ xv (i, j)

(1.3)

where integers i and j represent the abscissa and the ordinate, respectively. xh (i +1, j) is the horizontal axis status indicator, xv (i, j + 1) is the vertical axis status indicator, A11 , A12 , A21 , A22 , B1 , B2 , C1 , C2 are matrices with appropriate dimensions, and the boundary conditions of the system can be expressed as xh (0, j), xv (i, 0). (2) 2D-FM model 

x(i + 1, j + 1)  A1 x(i, j + 1) + A2 x(i + 1, j) + B 1 u(i, ¯ j + 1) + B 2 u(i ¯ + 1, j) y(i, j)  C x(i, j) + D u(i, ¯ j) (1.4)

where, integers i and j represent the abscissa and the ordinate, boundary conditions are x(0, j), x(i, 0), i, j  0, 1, 2, . . ., x(i, j) ∈ R n is in coordinates (i, j) of the local state variable, u(i, ¯ j) ∈ R m is the input variable; y(i, j) ∈ R l is the output variable, and A1 , A2 , B 1 , B 2 , C, D are matrices with appropriate dimensions. In addition, time delay is also one of the difficult aspects in batch process control, and its impact on batch processes should not be underestimated. Time delay is the

4

1 Introduction Fault Controller

Fault AT

Actuator

Controlled

Fault Sensor

Fig. 1.1 Passive fault-tolerant control structure diagram

main cause of system response delay and the extended running time. It not only damages the control performance of the system, but also increases the difficulty of system controller design. Since low cost, low energy consumption, and highefficiency production are the commercial pursuits, how to effectively deal with the time delay problem of batch process has become a hot topic in this field [155–161]. However, most of the above research results are for constant time delay, and research results on interval time-varying delays are rare [162–165]. As the structure of the system becomes more complex, the possibility of failures increases. The faults of the system can be divided into three types: actuator fault, sensor fault, and system internal fault, among which the actuator fault is the most serious. Once an actuator failure occurs in the system, it will not only affect the control performance of the system and the production efficiency of the product, but even pose a threat to the safety of life and property. Therefore, improving the system’s ability to respond to sudden failures and improving the control performance of the system in a faulty environment have become a very pressing issue in batch production processes. Fault-tolerant control is a common way to solve this problem. It enables a system that has failed to operate stably while meeting certain performance metrics. Classic fault-tolerant control mainly includes passive fault-tolerant control and active faulttolerant control. Passive fault-tolerant control means that when the system fails, the controller’s control structure and gain parameters are not changed, and the robust control technology is used to be insensitive to certain faults, such that the faulty system can continue under the original performance indicators. Stable operation can be “changed invariably,” and its structural framework is shown in Fig. 1.1. Active fault-tolerant control is essentially different from passive fault-tolerant control. After the system fails, the active fault-tolerant control can re-adjust the controller structure or change the controller parameters according to the results of fault diagnosis and detection to ensure that the system can still maintain stable operation after the fault occurs. The structural framework is shown in Fig. 1.2. Due to the complexity of the production process itself, batch processes are much more affected by the failure than continuous processes. As early as 1982, the famous scholar Wensley pointed out the importance and urgency of the study on fault-tolerant control in batch processes [166]. However, subject to the complexity of this issue

1.2 Current Status of Batch Process Control

5

Sensor r (t )

Controller

u (t )

Actuator

Reconfiguration / reconfiguration of the controller

Controlled

y (t )

Fault diagnosis and detection

Fig. 1.2 Active fault-tolerant control structure diagram

and the immature supporting technology, it was not until 2006 that relevant research results appeared [167]. With the wide application of ILC and the emergence of various batch process fault diagnosis methods, the problem of fault-tolerant control of batch processes has gradually gained attention, and a series of research results have emerged [58, 74, 168–175]. However, most of these research results are directed to batch processes with interference, and the results of fault-tolerant control for time delay are few. Most of the above studies on batch processes are directed to single-phase processes. In contrast, the results of multi-phase processes are relatively scarce [176–179]. For multi-phase characteristics, a small number of research results clearly do not meet the actual needs of the industrial market. Therefore, further analysis of the multi-phase batch process remains to be explored. In addition, compared with linear systems, nonlinear systems have always been a difficult and hot topic in the control field, because there are many unique dynamic characteristics in nonlinear models. These characteristics constitute a highly nonlinear batch process production model. It is very difficult, and it increases the difficulty of studying nonlinear systems. Due to the complexity of the nonlinear system itself and the immature development, research results on the nonlinear control of batch processes are few [180–189]. With the research progress, the authors find that the current robust iterative learning control method cannot solve the problem of system state deviation under the influence of interference and faults, whether it is the normal process or the faulty system of batch process; that is, the same control law is used from the beginning to the end. As time goes by, the deviation of the system state will increase. In addition, for the control law design and system output, the literature has rarely considered the constrained problem [190–193], but in the actual production process, the constrained problem cannot be ignored. In order to solve the above problems, model predictive control methods are widely used [194–199]. At present, most predictive control methods on batch processes are one dimensional; that is, only the time direction or batch direction is considered. However, only considering the time direction such that each batch is simply repeated leads to the fact that the control performance cannot be improved as the batch goes [200–208]; only considering the batch direction cannot cope with some of the control

6

1 Introduction Ch.1

Part.

Part.

Ch.2

Ch.3

Ch.4

Ch.5

Ch.6

Ch.7

Ch.8

Ch.9

Ch.10

Fig. 1.3 Relation among chapters

issues of batch process, such as initial value uncertainty [209]. It can be seen from the research of [150] that compared with corresponding controllers designed using the one-dimensional system model, the control performance on the two-dimensional system model is obviously better. However, in the face of the repeated nature, time-varying and complex reaction mechanism of batch processes, predictive control still shows certain limitations in its control performance. This requires a combination of other control algorithms to maximize the benefits of predictive control. Iterative learning control has obvious advantages in dealing with systems with repeatability and high precision requirements for tracking trajectories. Combining it with predictive control can make the control functions of the two interact with each other [210–212]. In summary, batch processes have become more and more important in modern industrial production by virtue of its own advantages. With the increasing complexity of production processes and in order to improve the stability of batch process operation and the control performance of the system, it is imperative to solve the following issues: (1) (2) (3) (4)

Control of batch processes with time-varying delays; Multi-phase control of batch process; Batch process control with constraints; Nonlinear system control.

1.3 The Main Content of This Book It can be seen from the above analysis that the research on batch processes still needs to be further studied. Accordingly, the specific structure of the book is arranged as shown in Fig. 1.3. The book is divided into ten chapters. The first chapter firstly describes the practical background and research significance of this topic and explains the related characteristics of batch processes and the difficulty for control technologies. Secondly, the basic system models of batch processes are introduced, and the differences of each model and the difficulty in research are pointed out. Then, the research topics

1.3 The Main Content of This Book

7

and related theory and research methods are introduced, which paves the way for the subsequent contents of the book. Subsequent contents are divided into two parts according to whether the system has faults. Part I mainly studies trouble-free batch process control, and Part II mainly studies fault-tolerant control of batch process. The contents of each chapter of the book are both related to and different from one another. For example, for Chaps. 2 and 3, although the former is a linear system and the latter is a nonlinear system, some control concepts in a linear system can also be applied to nonlinear systems; Chap. 4 studied the control problem of the multi-phase batch process. On the basis of the previous chapters, Chap. 5 not only considers the stable operation of the system, but also hopes that the system can maintain good control performance with stable operation, and thus studies the maintenance control problem of the batch process; Chap. 6 is different from the previous ones. It adopts the model predictive control method when the batch process contains constraints. Chaps. 7, 8, 9, and 10 are based on Chaps. 2, 3, 4, and 6, respectively, and study the control problem under actuator failures. This book solves some difficult problems in the current batch process control. It can be used as the guiding material for corresponding engineering control problems and can also provide some help for engineers and technicians. Some parts of the material in this book was published in relevant journals by the authors several years ago and included here for relevance and completeness. This includes: Section 2.2 of Chap. 2, reprinted from Journal of Process Control, volume 21, Wang L, Mo S, Zhou D, Gao F, “Robust design of feedback integrated with iterative learning control for batch processes with uncertainties and interval time-varying delays”, 987–996, ©2011 Elsevier Ltd., with permission from Elsevier. Section 2.3 of Chap. 2, reprinted with permission from Control Engineering Practice, volume 21, Wang L, Mo S, Qu H, Zhou D, Gao F. “H∞ design of 2D controller for batch processes with uncertainties and interval time-varying delays”, 1321–1333, ©2013 Elsevier Ltd., with permission from Elsevier. Section 2.4 of Chap. 2, reprinted from Journal of Process Control, volume 23, Wang L, Mo S, Zhou D, Gao F, Chen X. “Delay-range-dependent robust 2D iterative learning control for batch processes with state delay and uncertainties”, 715–730, ©2013 Elsevier Ltd., with permission from Elsevier. Section 3.2 of Chap. 3, reprinted with permission from Industrial & Engineering Chemistry Research, volume 56, Wang L, Zhu C, Yu J, Li P, Zhang R, Gao F. “Fuzzy iterative learning control for batch processes with interval time-varying delays”, ©2017 American Chemical Society. Section 5.2 of Chap. 5, reprinted with permission from AICHE Journal, volume 59, Wang L, Mo S, Zhou D, Gao F, Chen X, “Delay-range-dependent guaranteed cost control for batch processes with state delay”, 2033-2045, ©2013 John Wiley and Sons. Section 7.2 of Chap. 7, reprinted from Journal of Process Control, volume 21, Wang L, Mo S, Zhou D, Gao F, Chen X, “Robust delay dependent iterative learning fault-tolerant control for batch processes with state delay and actuator failures”, 1273–1286, ©2012 Elsevier Ltd., with permission from Elsevier.

8

1 Introduction

Section 7.3 of Chap. 7, reprinted with permission from Industrial & Engineering Chemistry Research, volume 52, Wang L, Mo S, Zhou D, Gao F, Chen X, “Delayrange-dependent method for iterative learning fault-tolerant guaranteed cost control for batch processes”, 2661–2671, ©2013 American Chemical Society. Section 8.3 of Chap. 8, reprinted with permission from Optimal Control Applications and Methods, volume 0, Wang L, Li B, Yu J, Zhang R, Gao F. “Design of fuzzy iterative learning fault-tolerant control for batch processes with time-varying delays”, ©2018 John Wiley and Sons. Section 9.2 of Chap. 9, reprinted with permission from Industrial & Engineering Chemistry Research, volume 56, Wang L, Sun L, Yu J, Zhang R, Gao F, “Robust iterative learning fault-tolerant control for multiphase batch processes with uncertainties”, ©2017 American Chemical Society. Section 9.3 of Chap. 9, reprinted from Journal of Process Control, volume 69, Shen Y, Wang L, Yu J, Zhang R, Gao F, “A hybrid 2D fault-tolerant controller design for multi-phase batch processes with time delay”, 138–157, ©2018 Elsevier Ltd., with permission from Elsevier. Section 9.4 of Chap. 9, reprinted with permission from Industrial & Engineering Chemistry Research, volume 57, Wang L, Liu B, Yu J, Li P, Zhang R, Gao F. “Delay-range-dependent-based hybrid iterative learning fault-tolerant guaranteed cost control for multiphase batch processes”, ©2018 American Chemical Society.

References 1. Xiong, Z., Zhang, J.: Modelling and optimal control of fed-batch processes using a novel control affine feedforward neural network. Neurocomputing 61(2), 317–337 (2004) 2. Stigter, J., Keesman, K.: Optimal parametric sensitivity control of a fed-batch reactor. Automatica 40(8), 1459–1464 (2004) 3. Xiong, Z., Jie, Z.: Optimal control of fed-batch processes based on multiple neural networks. Appl. Intell. 22(2), 149–161 (2005) 4. Peroni, C., Kaisare, N., Lee, J.: Optimal control of a fed-batch bioreactor using simulationbased approximate dynamic programming. IEEE Trans. Control Syst. Technol. 13(5), 786–790 (2005) 5. Gadkar, K., Mahadevan, R., Iii, F.: Optimal genetic manipulations in batch bioreactor control. Automatica 42(10), 1723–1733 (2006) 6. Koseoglu, G., Ileri, R.: Decolorization of dying textile wastewater by advanced activated sludge in a sequencing batch reactor. Fresenius Environ. Bull. 15(9B), 1111–1114 (2006) 7. Xiong, Z., Jie, Z., Jin, D.: Optimal iterative learning control for batch processes based on linear time-varying perturbation model. Chin. J. Chem. Eng. 16(2), 235–240 (2008) 8. Xiong, Z., Xu, Y., Zhang, J., et al.: Batch-to-batch control of fed-batch processes using controlaffine feedforward neural network. Neural Comput. Appl. 17(4), 425–432 (2008) 9. Wang, Y., Gao, F., Iii, F.J.D.: Survey on iterative learning control, repetitive control, and run-to-run control. J. Process Control 19(10), 1589–1600 (2009) 10. Zhang, R., Wang, S.: Support vector machine based predictive functional control design for output temperature of coking furnace. J. Process Control 18(5), 439–448 (2008) 11. Christensen, A., Gurol, M., Garoma, T.: Treatment of persistent organic compounds by integrated advanced oxidation processes and sequential batch reactor. Water Res. 43(16), 3910–3921 (2009)

References

9

12. Singhal, S., Elkhatib, B., Stuber, J., et al.: Characterization of wet batch cleaning process in advanced semiconductor manufacturing. IEEE Trans. Semicond. Manuf. 22(3), 399–408 (2009) 13. Rapaport, A., Dochain, D.: Minimal time control of fed-batch processes with growth functions having several maxima. IEEE Trans. Autom. Control 43(6), 96–101 (2010) 14. Wang, Y., Liu, T., Zhao, Z.: Advanced PI control with simple learning set-point design: application on batch processes and robust stability analysis. Chem. Eng. Sci. 71(13), 153–165 (2012) 15. Zhang, R., Tao, J., Gao, F.: A new approach of Takagi-Sugeno fuzzy modeling using improved GA optimization for oxygen content in a coke furnace. Ind. Eng. Chem. Res. 55(22), 6465–6474 (2016) 16. Yin, S., Ding, S., Abandan Sari, A., et al.: Data-driven monitoring for stochastic systems and its application on batch process. Int. J. Syst. Sci. 44(7), 1366–1376 (2013) 17. Niu, D., Jia, M., Wang, F., et al.: Optimization of nosiheptide fed-batch fermentation process based on hybrid model. Ind. Eng. Chem. Res. 52(9), 3373–3380 (2013) 18. Zhang, R., Lu, J., Qu, H., et al.: State space model predictive fault-tolerant control for batch processes with partial actuator failure. J. Process Control 24(5), 613–620 (2014) 19. Ye, J., Xu, H., Feng, E., et al.: Optimization of a fed-batch bioreactor for 1,3-propanediol production using hybrid nonlinear optimal control. J. Process Control 24(10), 1556–1569 (2014) 20. Abburi, T., Narasimhan, S.: Optimal sensor scheduling in batch processes using convex relaxations and tchebycheff systems theory. IEEE Trans. Autom. Control 59(11), 2978–2983 (2014) 21. Rodríguez, J., Ramírez, H., Gajardo, P., et al.: Optimality of affine control system of several species in competition on a sequential batch reactor. Int. J. Control 87(9), 1877–1885 (2014) 22. Hosen, M., Hussain, M., Mjalli, F., et al.: Performance analysis of three advanced controllers for polymerization batch reactor: an experimental investigation. Chem. Eng. Res. Des. 92(5), 903–916 (2014) 23. Zou, Q., Jin, Q., Zhang, R.: Design of fractional order predictive functional control for fractional industrial processes. Chemometr. Intell. Lab. Syst. 152, 34–41 (2016) 24. Bao, B., Yin, H., Feng, E.: Computation of impulsive optimal control for 1,3-PD fed-batch culture. J. Process Control 34, 49–55 (2015) 25. Bayen, T., Mairet, F., Mazade, M.: Optimal feeding strategy for the minimal time problem of a fed-batch bioreactor with mortality rate. Optim. Control. Appl. Methods 36(1), 77–92 (2015) 26. Zhang, R., Zou, H., Xue, A., Gao, F.: GA based predictive functional control for batch processes under actuator faults. Chemometr. Intell. Lab. Syst. 137, 67–73 (2014) 27. Kern, R., Shastri, Y.: Advanced control with parameter estimation of batch transesterification reactor. J. Process Control 33, 127–139 (2015) 28. Qin, Y., Zhao, C., Gao, F.: An iterative two-step sequential phase partition (ITSPP) method for batch process modeling and online monitoring. AIChE J. 62(7), 2358–2373 (2016) 29. Holmqvist, A., Magnusson, F.: Open-loop optimal control of batch chromatographic separation processes using direct collocation. J. Process Control 46, 55–74 (2016) 30. Zhang, S., Zhao, C., Gao, F.: Two-directional concurrent strategy of mode identification and sequential phase division for multimode and multiphase batch process monitoring with uneven lengths. Chem. Eng. Sci. 178, 104–117 (2018) 31. Cao, Z., Lu, J., Zhang, R., et al.: Online average-based system modelling method for batch process. Comput. Chem. Eng. 108, 128–138 (2018) 32. Jaramillo, F., Orchard, M., Muãoz, C, et al.: Advanced strategies to improve nitrification process in sequencing batch reactors—a review. J. Environ. Manag. 218, 154–164 (2018) 33. Schumacher, J.: A framework for batch-operation analysis within the context of disturbance management. Comput. Chem. Eng. 24(2), 1175–1180 (2000) 34. Kookos, I.: Optimal operation of batch processes under uncertainty: A monte carlo simulationdeterministic optimization approach. Ind. Eng. Chem. Res. 42(26), 6815–6822 (2003)

10

1 Introduction

35. Zhang, R., Cao, Z., Bo, C., Li, P., Gao, F.: New PID controller design using extended nonminimal state space model based predictive functional control structure. Ind. Eng. Chem. Res. 53(8), 3283–3292 (2014) 36. Renard, F., Wouwer, A., Valentinotti, S., et al.: A practical robust control scheme for yeast fed-batch cultures—an experimental validation. J. Process Control 16(8), 855–864 (2006) 37. Dovžan, D., Škrjanc, I.: Predictive functional control based on an adaptive fuzzy model of a hybrid semi-batch reactor. Control. Eng. Pract. 18(8), 979–989 (2010) 38. Zhu, J., Gu, X., Gu, W.: Optimal multiperiod scheduling of multiproduct batch plants under demand uncertainty. Kybernetes 40(5/6), 871–882(12) (2011) 39. Mo, S., Wang, L., Yao, Y., et al.: Two-time dimensional dynamic matrix control for batch processes with convergence analysis against the 2D interval uncertainty. J. Process Control 22(5), 899–914 (2012) 40. Konakom, K., Kittisupakorn, P., Mujtaba, I.M.: Neural network-based controller design of a batch reactive distillation column under uncertainty. Asia-Pac. J. Chem. Eng. 7(3), 361–377 (2012) 41. Terwiesch, P., Agarwal, M.: Robust input policies for batch reactors under parametric uncertainty. Chem. Eng. Commun. 131(1), 33–52 (2012) 42. Li, J., Yang, T., Chiu, M.: An integrated iterative learning control strategy with model identification and dynamic R-parameter for batch processes. J. Process Control 23(9), 1332–1341 (2013) 43. Kadri, M.: Disturbance rejection in nonlinear uncertain systems using feedforward control. Arab. J. Sci. Eng. 38(9), 2439–2450 (2013) 44. Zhu, J., Gu, X., Gu, W.: Robust optimization approach for short-term scheduling of batch plants under demand uncertainty. Kybernetes 40(5/6), 860–870 (2013) 45. Benavides, T., Diwekar, U.: Studying various optimal control problems in biodiesel production in a batch reactor under uncertainty. Fuel 103(5), 585–592 (2013) 46. Lara-Cisneros, G., Femat, R., Dochain, D.: An extremum seeking approach via variablestructure control for fed-batch bioreactors with uncertain growth rate. J. Process Control 24(5), 663–671 (2014) 47. Pinto-Varela, T., Barbosa-Póvoa, A., Carvalho, A.: Sustainable batch process retrofit design under uncertainty—an integrated methodology. Comput. Chem. Eng. 102(12), 226–237 (2016) 48. Jia, L., Han, C., Chiu, M.: Dynamic R-parameter based integrated model predictive iterative learning control for batch processes. J. Process Control 49, 26–35 (2017) 49. Wang, Y., Zhao, D., Li, Y., et al.: Unbiased minimum variance fault and state estimation for linear discrete time-varying two-dimensional systems. IEEE Trans. Autom. Control 62(10), 5463–5469 (2017) 50. Huang, W., Fan, H., Qian, Y., et al.: Assessment and computation of the delay tolerability for batch reactors under uncertainty. Chem. Eng. Res. Des. 124, 74–84 (2017) 51. Huang, W., Yang, G., Qian, Y., et al.: Operation feasibility analysis of chemical batch reaction systems with production quality consideration under uncertainty. J. Chem. Eng. Jpn. 50(1), 45–52 (2017) 52. Hille, R, Mandur, J, Budman, H.: Robust batch-to-batch optimization in the presence of model-plant mismatch and input uncertainty. Aiche J 63(7), 2660–2670 (2017) 53. Oh, S., Lee, J.: Iterative learning control integrated with model predictive control for real-time disturbance rejection of batch processes. J. Chem. Eng. Jpn. 50(6), 415–421 (2017) 54. Aydin, E., Bonvin, D., Kai, S.: Computationally efficient NMPC for batch and semi-batch processes using parsimonious input parameterization. J. Process Control 66, 12–22 (2018) 55. Aydin, E., Bonvin, D., Kai, S.: NMPC using pontryagin’s minimum principle-application to a two-phase semi-batch hydroformylation reactor under uncertainty. Comput. Chem. Eng. 108, 47–56 (2018) 56. Lu, P., Chen, J., Xie, L.: ILC based economic optimization for batch processes using helpful disturbance information. Ind. Eng. Chem. Res. 57(10), 3717–3731 (2018)

References

11

57. Shen, Y., Wang, L., Yu, J., Zhang, R., Gao, F.: A hybrid 2D fault-tolerant controller design for multi-phase batch processes with time delay. J. Process Control 69, 138–157 (2018) 58. Wang, L., Liu, F., Yu, J., et al.: Iterative learning fault-tolerant control for injection molding processes against actuator faults. J. Process Control 59, 59–72 (2017) 59. Wang, L., Dong, W.: Optimal iterative learning fault-tolerant guaranteed cost control for batch processes in the 2D-FM model. Abstr. Appl. Anal. 5, 919–929 (2012) 60. Wang, L., Yu, J., Shi, J., et al.: Delay-range dependent H∞ control for uncertain 2D-delayed systems. Numer. Algebr. Control. Optim. 5(1), 11–23 (2015) 61. Wang, L., Yang, J., Yu, J., et al.: Iterative learning fault tolerant control of batch process based on T-S fuzzy model. Chin. J. Chem. Ind. 68(3), 1081–1089 (2017) 62. Pierri, F., Paviglianiti, G.: Observer-based actuator fault detection for chemical batch reactors: a comparison between nonlinear adaptive and H∞—based approaches. In: Mediterranean Conference on Control & Automation, 2007. MED ‘07. IEEE, pp. 1–6 (2007) 63. Liu, T., Gao, F.: A generalized relay identification method for time delay and non-minimum phase processes. Automatica 45(4), 1072–1079 (2009) 64. Aumi, S., Mhaskar, P.: Robust model predictive control & fault. In: American Control Conference (ACC), 2010. IEEE, pp. 4415–4420 (2010) 65. Liu, K., Fridman, E., Hetel, L.: Stability and L2L2-gain analysis of networked control systems under round-robin scheduling: a time-delay approach. Syst. Control Lett. 61(5), 666–675 (2012) 66. Zhang, R., Gan, L., Lu, J., et al.: New design of state space linear quadratic fault-tolerant tracking control for batch processes with partial actuator failure. Ind. Eng. Chem. Res. 52(46), 16294–16300 (2013) 67. Liu, C.: Modelling and parameter identification for a nonlinear time-delay system in microbial batch fermentation. Appl. Math. Model. 37(10–11), 6899–6908 (2013) 68. Zhang, R., Lu, R., Xue, A., et al.: Predictive functional control for linear systems under partial actuator faults and application on an injection molding batch process. Ind. Eng. Chem. Res. 53(2), 723–731 (2014) 69. Liu, C.: Optimal control of a switched autonomous system with time delay arising in fed-batch processes. Ima J. Appl. Math. 80(2), 569–584 (2015) 70. Ertiame, A., Yu, D., Yu, F., et al.: Robust fault diagnosis for an exothermic semi-batch polymerization reactor under open-loop. Syst. Sci. Control. Eng. 3(1), 14–23 (2015) 71. Zhang, R., Lu, R., Xue, A., et al.: New minmax linear quadratic fault-tolerant tracking control for batch processes. IEEE Trans. Autom. Control 61(10), 3045–3051 (2016) 72. Liu, C., Gong, Z., Teo, K., et al.: Multi-objective optimization of nonlinear switched timedelay systems in fed-batch process. Appl. Math. Model. 40(23–24), 10533–10548 (2016) 73. Tao, Y., Shen, D., Fang, M., et al.: Reliable H∞ control of discrete-time systems against random intermittent faults. Int. J. Syst. Sci. 47(10), 2290–2301 (2016) 74. Tao, H., Zou, W., Yang, H.: Robust dissipative iterative learning fault-tolerant control for multi-rate sampling batch process with actuator failure. Control. Theory Appl. 33(3), 329–335 (2016) 75. Liu, C., Gong, Z., Teo, K., et al.: Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process. Optim. Lett. 1–16 (2016) 76. Tao, H., Paszke, W., Rogers, E., et al.: Iterative learning fault-tolerant control for differential time-delay batch processes in finite frequency domains. J. Process Control 56, 112–128 (2017) 77. Renganathan, K.: Estimation based fault diagnosis and identification in sequential industrial batch processes modeled as Hybrid Petri nets. In: International Conference on Computing and Communications Technologies. IEEE, 2017, pp. 64–68 78. Zou, T., Wu, S., Zhang, R.: Improved state space model predictive fault-tolerant control for injection molding batch processes with partial actuator faults using GA optimization. ISA Trans. 73, 147–153 (2018) 79. Hwang, Y., Lee, S., Chang, H., et al.: Dissolved oxygen concentration regulation using autotuning proportional-integral-derivative controller in fermentation process. Biotechnol. Tech. 5(2), 85–90 (1991)

12

1 Introduction

80. Miklovicova, E., Latifi, M., M’Saad, M., et al.: PID adaptive control of the temperature in batch and semi-batch chemical reactors. Chem. Eng. Sci. 51(11), 3139–3144 (1996) 81. Huzmezan, M., Gough, W., Dumont, G., et al.: Time delay integrating systems: a challenge for process control industries. A practical solution. Control. Eng. Pract. 10(10), 1153–1161 (2002) 82. Chen, J., Huang, T.: Applied neural network to on-line updated PID controllers for nonlinear process control. J. Process Control 14(2), 211–230 (2004) 83. Yeo, Y., Kwon, T., Lee, K.: An energy effective PID tuning method for the control of polybutadiene latex reactor based on closed-loop identification. Korean J. Chem. Eng. 21(5), 935–941 (2004) 84. Hagenmeyer, V., Nohr, M.: Flatness-based two-degree-of-freedom control of industrial semibatch reactors using a new observation model for an extended Kalman filter approach. Int. J. Control 81(3), 428–438 (2008) 85. Altınten, A., Ketevanlio˘glu, F., Erdo˘gan, S., et al.: Self-tuning PID control of jacketed batch polystyrene reactor using genetic algorithm. Chem. Eng. J. 138(1), 490–497 (2008) 86. Altinten, A., Karakurt, S., Erdogan, S., et al.: Application of self-tuning PID control with genetic algorithm to a semi-batch polystyrene reactor. Indian J. Chem. Technol. 17(5), 356–365 (2010) 87. Roeva, O., Slavov, T.: PID controller tuning based on metaheuristic algorithms for bioprocess control. Biotechnol. Biotechnol. Equip. 26(5), 3267–3277 (2012) 88. Liu, T., Wang, X., Chen, J.: Robust PID based indirect-type iterative learning control for batch processes with time-varying uncertainties. J. Process Control 24(12), 95–106 (2014) 89. Guzmánguillén, R., Prieto, A., Moreno, I., et al.: Universal DIN PID process controller: handles batch and continuous processes (Product News). Environ. Toxicol. 30(3), 261–277 (2015) 90. Chan, L., Chen, T., Chen, J.: PID based nonlinear processes control model uncertainty improvement by using Gaussian process model. J. Process Control 42, 77–89 (2016) 91. Marchal-Brassely, S., Villermaux, J., Houzelot, J., et al.: Optimal operation of a semi-batch reactor by self-adaptive models for temperature and feed-rate profiles. Chem. Eng. Sci. 47(9–11), 2445–2450 (1992) 92. Defaye, G., Regnier, N., Chabanon, J., et al.: Adaptive—predictive temperature control of semi-batch reactors. Chem. Eng. Sci. 48(19), 3373–3382 (1993) 93. Narendra, K.: Comparison of linear, nonlinear and neural-network-based adaptive controllers for a class of fed-batch fermentation processes. Automatica 31(6), 817–840 (1995) 94. Van, I., Bastin, G.: Optimal adaptive control of fed-batch fermentation processes with multiple substrates. Control. Eng. Pract. 3(7), 939–954 (1995) 95. Chiotti, O., Salomone, H., Iribarren, O.: Batch plants with adaptive operating policies. Comput. Chem. Eng. 20(20), 1241–1256 (1996) 96. Pu, L., Wozny, G.: Tracking the predefined optimal policies for multiple-fraction batch distillation by using adaptive control. Comput. Chem. Eng. 25(1), 97–107 (2001) 97. And, F., Macgregor, J.: Within-batch and batch-to-batch inferential-adaptive control of semibatch reactors: a partial least squares approach. Ind. Eng. Chem. Res. 42(14), 3334–3345 (2003) 98. Holmberg, U., Bonvin, D., Cannizzaro, C.: Optimal operation of fed-batch fermentations via adaptive control of overflow metabolite. Control Eng. Pract. 11(6), 665–674 (2003) 99. Titica, M., Dochain, D., Guay, M.: Adaptive extremum seeking control of fed-batch bioreactors. Eur. J. Control. 9(6), 618–631 (2003) 100. Khazraee, S., Jahanmiri, A.: Composition estimation of reactive batch distillation by using adaptive neuro-fuzzy inference system. Chin. J. Chem. Eng. 18(4), 703–710 (2010) 101. Khazraee, S., Jahanmiri, A., Ghorayshi, S.: Model reduction and optimization of reactive batch distillation based on the adaptive neuro-fuzzy inference system and differential evolution. Neural Comput. Appl. 20(2), 239–248 (2011) 102. Chen, C., Peng, S.: A simple adaptive control strategy for temperature trajectory tracking in batch processes. Can. J. Chem. Eng. 45(15), 644–649 (2012)

References

13

103. Zanella, A.: Adaptive batch resolution algorithm with deferred feedback for wireless systems. IEEE Trans. Wireless Commun. 11(10), 3528–3539 (2012) 104. Claes, J.: Optimal adaptive control of the fed-batch baker’s yeast fermentation process. J. Process Control 23(8), 1159–1168 (2013) 105. Su, Q., Hermanto, M., Braatz, R., et al.: Just-in-time-learning based extended prediction self-adaptive control for batch processes. J. Process Control 43, 1–9 (2016) 106. Li, Y., Wang, N., Shi, J., et al.: Adaptive batch normalization for practical domain adaptation. Pattern Recogn. 80, 109–117 (2018) 107. Arimoto, S., Kawamura, S., Miyazaki, F.: Bettering operation of robots by learning. J. Robotic Syst. 1(2), 123–140 (1984) 108. Gao, F., Yang, Y., Shao, C.: Robust iterative learning control with applications to injection molding process. Chem. Eng. Sci. 56(24), 7025–7034 (2001) 109. Lee, K., Lee, J.: Iterative learning control-based batch process control technique for integrated control of end product properties and transient profiles of process variables. J. Process Control 13(7), 607–621 (2003) 110. And, Z., Jie, Z.: Product quality trajectory tracking in batch processes using iterative learning control based on time-varying perturbation models. Ind. Eng. Chem. Res. 42(26), 6802–6814 (2003) 111. Xiong, Z., Zhang, J., Wang, X., et al.: Tracking control for batch processes through integrating batch-to-batch iterative learning control and within-batch on-line control. Ind. Eng. Chem. Res. 44(11), 3983–3992 (2005) 112. Xiong, Z., Jie, Z.: A batch-to-batch iterative optimal control strategy based on recurrent neural network models. J. Process Control 15(1), 11–21 (2005) 113. And, F., Macgregor, J.: Iterative learning control for final batch product quality using partial least squares models. Ind. Eng. Chem. Res. 44(24), 9146–9155 (2005) 114. Lee, J., Lee, K.: Iterative learning control applied to batch processes: an overview. Control Eng. Pract. 15(10), 1306–1318 (2007) 115. Chen, J., Lin, K.: Batch-to-batch iterative learning control and within-batch on-line control for end-point qualities using MPLS-based dEWMA. Chem. Eng. Sci. 63(4), 977–990 (2008) 116. Cho, W., Edgar, T., Lee, J.: Iterative learning dual-mode control of exothermic batch reactors. Control Eng. Pract. 16(10), 1244–1249 (2008) 117. Chen, J., ChihChiang, C., Munoz, C., et al.: Iterative Learning Control for Processes with Uncertain Time Delay Using Minimum Entropy. IEEE, pp. 704–709 (2010) 118. Hladowski, L., Galkowski, K., Cai, Z., et al.: Experimentally supported 2D systems based iterative learning control law design for error convergence and performance. Control Eng. Pract. 18(4), 339–348 (2010) 119. Huang, D., Xu, J.: Steady-state iterative learning control for a class of nonlinear PDE processes. J. Process Control 21(8), 1155–1163 (2011) 120. Jewaratnam, J., Zhang, J., Morris, J., et al.: Batch-to-Batch Iterative Learning Control Using Linearised Models with Adaptive Model Updating. IEEE, pp. 271–276 (2012) 121. Ruan, X., Bien, Z.: Pulse compensation for PD-type iterative learning control against initial state shift. Int. J. Syst. Sci. 43(11), 2172–2184 (2012) 122. Jia, L., Shi, J., Chiu, M.S.: Integrated neuro-fuzzy model and dynamic R-parameter based quadratic criterion-iterative learning control for batch process. Neurocomputing 98(18), 24–33 (2012) 123. Paszke, W., Rogers, E., Gałkowski, K., et al.: Robust finite frequency range iterative learning control design and experimental verification. Control Eng. Pract. 21(10), 1310–1320 (2013) 124. Sanzida, N., Nagy, Z.: Iterative learning control for the systematic design of supersaturation controlled batch cooling crystallisation processes. Comput. Chem. Eng. 59(10), 111–121 (2013) 125. Farasat, E., Huang, B.: Deterministic versus stochastic performance assessment of iterative learning control for batch processes. Aiche J 59(2), 457–464 (2013) 126. Shi, J., Zhou, H., Cao, Z., et al.: A design method for indirect iterative learning control based on two-dimensional generalized predictive control algorithm. J. Process Control 24(10), 1527–1537 (2014)

14

1 Introduction

127. Wang, J., Wang, Y., Wang, W., et al.: Adaptive iterative learning control based on unsatisfied strategy applied in batch process. J. Central South Univ. 46(4), 1318–1325 (2015) 128. Wang, Y., Chien, C., Chi, R., et al.: A fuzzy-neural adaptive terminal iterative learning control for fed-batch fermentation processes. Int. J. Fuzzy Syst. 17(3), 423–433 (2015) 129. Oh, S., Lee, J.: Stochastic iterative learning control for discrete linear time-invariant system with batch-varying reference trajectories. J. Process Control 36, 64–78 (2015) 130. Xuan, Y., Xiao, E., et al.: Robustness of reinforced gradient-type iterative learning control for batch processes with Gaussian noise. Chin. J. Chem. Eng. 24(5), 623–629 (2016) 131. Bolder, J., Oomen, T.: Inferential iterative learning control: A 2D-system approach. Automatica 71, 247–253 (2016) 132. Cui-Mei, B., Yang, L., Huang, Q., et al.: 2D multi-model general predictive iterative learning control for semi-batch reactor with multiple reactions. J. Central South Univ. (Engl. Version) 24(11), 2613–2623 (2017) 133. Wang, L., Shen, Y., Yu, J., et al.: Robust iterative learning control for multi-phase batch processes: an average dwell-time method with 2D convergence indexes. Int. J. Syst. Sci. 49(2), 324–343 (2018) 134. Amann, N., Owens, D.: Non-minimum phase plants in iterative learning control. In: International Conference on Intelligent Systems Engineering. IET, pp. 107–112 (1994) 135. Roh, C., Lee, M., Chung, M.: ILC for non-minimum phase system. Int. J. Syst. Sci. 27(4), 419–424 (1996) 136. Park, K., Bien, Z., Hwang, D.: Design of an Iterative Learning Controller for a Class of Linear Dynamic Systems With Time-Delay and Initial State Error. Iterative Learning Control. Springer, US (1998) 137. Hideg, L.: Time delays in iterative learning control schemes. In: IEEE International Symposium on Intelligent Control. IEEE, pp. 215–220 (1995) 138. Lee, K., Bang, S., Chang, K.: Feedback-assisted iterative learning control based on inverse process model. J. Process Control 4(4), 77–89 (1994) 139. Chin, I., Qin, S., Lee, K., et al.: A two-phase iterative learning control technique combined with real-time feedback for independent disturbance rejection. Automatica 40(11), 1913–1922 (2004) 140. Shi, J., Gao, F., Wu, T.: Robust design of integrated feedback and iterative learning control of a batch process based on a 2D Roesser system. J. Process Control 15(8), 907–924 (2005) 141. Shi, J., Gao, F., Wu, T.: Integrated design and structure analysis of robust iterative learning control system based on a two-dimensional model. Ind. Eng. Chem. Res. 44(21), 8095–8105 (2005) 142. Shi, J., Gao, F., Wu, T.: A Robust iterative learning control design for batch processes with uncertain perturbations and initialization. AICHE J. 52(6), 2171–2187 (2006) 143. Shi, J., Gao, F., Wu, T.: From two-dimensional linear quadratic optimal control to iterative learning control. Paper 2. Iterative learning controls for batch processes. Ind. Eng. Chem. Res. 45(13), 4617–4628 (2006) 144. Shi, J., Gao, F., Wu, T.: From two-dimensional linear quadratic optimal control to iterative learning control. Paper 1. Two-dimensional linear quadratic optimal controls and system analysis. Ind. Eng. Chem. Res. 45(13), 4603–4616 (2006) 145. Wang, Y., Zhou, D., Gao, F.: Iterative learning model predictive control for multi-phase batch processes. J. Process Control 18(6), 543–557 (2008) 146. Wang, L., Mo, S., Zhou, D., et al.: Robust design of feedback integrated with iterative learning control for batch processes with uncertainties and interval time-varying delays. J. Process Control 21(7), 987–996 (2011) 147. Wang, Y., Yang, Y., Zhao, Z.: Robust stability analysis for an enhanced ILC-based PI controller. J. Process Control 23(2), 201–214 (2013) 148. Yu, X., Xiong, Z., Huang, D., et al.: Model-based iterative learning control for batch processes using generalized hinging hyperplanes. Ind. Eng. Chem. Res. 52(4), 1627–1634 (2013) 149. Shen, D., Wang, Y.: Survey on stochastic iterative learning control. J. Process Control 24(12), 64–77 (2014)

References

15

150. Chen, C., Xiong, Z., et al.: Design and analysis of integrated predictive iterative learning control for batch process based on two-dimensional system theory. Chin. J. Chem. Eng. (Eng. version) 22(7), 762–768 (2014) 151. Cao, Z., Zhang, R., Yang, Y., et al.: Discrete-time robust iterative learning kalman filtering for repetitive processes. IEEE Trans. Autom. Control 61(1), 270–275 (2015) 152. Shen, D., Zhang, W., Wang, Y., et al.: On almost sure and mean square convergence of P-type ILC under randomly varying iteration lengths. Automatica, 63(C), 359–365 (2016) 153. Roesser, R.: A discrete state-space model for linear image processing. IEEE Trans. Autom. Control 20(1), 1–10 (1975) 154. Fornasini, E., Marchesini, G.: State-space realization theory of two-dimensional filters. IEEE Trans. Autom. Control 21(4), 484–492 (1976) 155. Xu, J., Hu, Q., Tong, H., et al.: Iterative learning control with Smith time delay compensator for batch processes. J. Shenyang Inst. Chem. Technol. 11(3), 321–328 (2003) 156. Tay, A., Weng, K., Deng, J., et al.: Control of photoresist film thickness: Iterative feedback tuning approach. Comput. Chem. Eng. 30(3), 572–579 (2006) 157. Cai, Z., Freeman, C., Lewin, P., et al.: Iterative learning control for a non-minimum phase plant based on a reference shift algorithm. Control. Eng. Pract. 16(6), 633–643 (2008) 158. Tan, K., Shao, Z., Huang, S., et al.: A new repetitive control for LTI systems with input delay. J. Process Control 19(4), 711–716 (2009) 159. Liu, T., Gao, F., Wang, Y.: IMC-based iterative learning control for batch processes with uncertain time delay. J. Process Control 20(2), 173–180 (2010) 160. Liu, C., Gong, Z., Feng, E.: Optimal control for a nonlinear time-delay system in fed-batch fermentation. Pac. J. Optim. 9(4), 595–612 (2013) 161. Liu, C.: Sensitivity analysis and parameter identification for a nonlinear time-delay system in microbial fed-batch process. Appl. Math. Model. 38(4), 1449–1463 (2014) 162. Liu, T., Gao, F.: Robust two-dimensional iterative learning control for batch processes with state delay and time-varying uncertainties. Chem. Eng. Sci. 65(23), 6134–6144 (2010) 163. Wang, L., Mo, S., Qu, H., et al.: H∞, design of 2D controller for batch processes with uncertainties and interval time-varying delays. Control Eng. Pract. 21(10), 1321–1333 (2013) 164. Wang, L., Mo, S., Zhou, D., et al.: Delay-range-dependent guaranteed cost control for batch processes with state delay. AICHE J. 59(6), 2033–2045 (2013) 165. Wang, L., Mo, S., Zhou, D., et al.: Delay-range-dependent robust 2D iterative learning control for batch processes with state delay and uncertainties. J. Process Control 23(5), 715–730 (2013) 166. Wensley, J.: Fault-tolerant control for batch processes. Intech 29(10), 69–71 (1982) 167. Wang, Y., Shi, J., Zhou, D., et al.: Iterative learning fault-tolerant control for batch processes. Ind. Eng. Chem. Res. 45(26), 9050–9060 (2006) 168. Wang, Y., Zhou, D., Gao, F.: Iterative learning reliable control of batch processes with sensor faults. Chem. Eng. Sci. 63(4), 1039–1051 (2008) 169. Wang, L., Mo, S., Zhou, D., et al.: Robust delay dependent iterative learning fault-tolerant control for batch processes with state delay and actuator failures. J. Process Control 22(7), 1273–1286 (2012) 170. Liu, T., Wang, Y.: A synthetic approach for robust constrained iterative learning control of piecewise affine batch processes. Automatica 48(11), 2762–2775 (2012) 171. Wang, L., Mo, S., Zhou, D., et al.: Delay-range-dependent method for iterative learning faulttolerant guaranteed cost control for batch processes. Ind. Eng. Chem. Res. 52(7), 2661–2671 (2013) 172. Wang, L., Chen, X., Gao, F.: An LMI method to robust iterative learning fault-tolerant guaranteed cost control for batch processes. Chin. J. Chem. Eng. 21(4), 401–411 (2013) 173. Wang, L., Dong, W.: Optimal iterative learning fault-tolerant guaranteed cost control for batch processes in the 2D-FM model. Abstr. Appl. Anal. 1–20 (2012) 174. Wang, L., Zhou, D., Zhu, C., et al.: Composite iterative learning fault-tolerant guaranteed cost controller design for batch process. J. Shanghai Jiaotong Univ. 49(6), 743–750 (2015)

16

1 Introduction

175. Gao, M., Sheng, L., Zhou, D., et al.: Iterative learning fault-tolerant control for networked batch processes with multi-rate sampling and quantization effects. Ind. Eng. Chem. Res. 56(9), 2515–2525 (2017) 176. Wang, L., He, X., Zhou, D.: Average dwell time-based optimal iterative learning control for multi-phase batch processes. J. Process Control 40, 1–12 (2016) 177. Wang, L., Shen, Y., Li, B., et al.: Hybrid iterative learning fault-tolerant guaranteed cost control design for multi-phase batch processes. Can. J. Chem. Eng. 96(2), 521–530 (2017) 178. Wang, L., Sun, L., Yu, J., et al.: Robust iterative learning fault-tolerant control for multiphase batch processes with uncertainties. Ind. Eng. Chem. Res. 56(36), 10099–10109 (2017) 179. Wang, L., Liu, B., Yu, J., et al.: Delay-range-dependent-based hybrid iterative learning faulttolerant guaranteed cost control for multiphase batch processes. Ind. Eng. Chem. Res. 57(8), 2932–2944 (2018) 180. Zhang, H., Li, Q.: Fuzzy programming based modeling of production scheduling of batch processes. Eng. Sci. 6(10), 65–70 (2004) 181. Fu, C., Wang, S., Wang, J.: Region fuzzy control for batch processes part 2. Feed timing prediction and control for an antibiotic fermentation production process. Int. J. Syst. Sci. 21(10), 1911–1921 (2007) 182. Ashoori, A., Moshiri, B., Khaki-Sedigh, A., et al.: Optimal control of a nonlinear fed-batch fermentation process using model predictive approach. J. Process Control 19(7), 1162–1173 (2009) 183. Kathel, P., Jana, A.: Dynamic simulation and nonlinear control of a rigorous batch reactive distillation. ISA Trans. 49(1), 130–137 (2010) 184. Yin, C., Xu, J., Hou, Z.: A high-order internal model based iterative learning control scheme for nonlinear systems with time-iteration-varying parameters. IEEE Trans. Autom. Control 55(11), 2665–2670 (2010) 185. Cosenza, B., Galluzzo, M.: Nonlinear fuzzy control of a fed-batch reactor for penicillin production. Comput. Chem. Eng. 36(1), 273–281 (2012) 186. Shen, D., Wang, Y.: ILC for networked nonlinear systems with unknown control direction through random Lossy channel. Syst. Control Lett. 77, 30–39 (2015) 187. Peng, K., Zhang, K., You, B., et al.: A quality-based nonlinear fault diagnosis framework focusing on industrial multimode batch processes. IEEE Trans. Industr. Electron. 63(4), 2615–2624 (2016) 188. Wang, L., Zhu, C., Yu, J., et al.: Fuzzy iterative learning control for batch processes with interval time-varying delays. Ind. Eng. Chem. Res. 56(14), 3993–4001 (2017) 189. Wang, L., Li, B., Yu, J., et al.: Design of fuzzy iterative learning fault-tolerant control for batch processes with time-varying delays. Optim. Control. Appl. Methods (2018). (in press) 190. Bing, C., Owens, D.H.: Iterative learning control for constrained linear systems. Int. J. Control 83(7), 1397–1413 (2010) 191. Zhang, Y., Fan, Y., Zhang, P.: Combining kernel partial least-squares modeling and iterative learning control for the batch-to-batch optimization of constrained nonlinear processes. Ind. Eng. Chem. Res. 49(16), 7470–7477 (2010) 192. Yuan, J., Liu, C., Zhang, X., et al.: Optimal control of a batch fermentation process with nonlinear time-delay and free terminal time and cost sensitivity constraint. J. Process Control 44(8), 41–52 (2016) 193. Oh, S., Lee, J.: Iterative learning model predictive control for constrained multivariable control of batch processes. Comput. Chem. Eng. 93, 284–292 (2016) 194. Balasubramhanya, L., Iii, F.: Nonlinear model-based control of a batch reactive distillation column. J. Process Control 10(2), 209–218 (2000) 195. Seki, H., Ogawa, M., Ooyama, S., et al.: Industrial application of a nonlinear model predictive control to polymerization reactors. Control Eng. Pract. 9(8), 819–828 (2001) 196. Nagy, Z., Braatz, R.: Robust nonlinear model predictive control of batch processes. AICHE J. 49(7), 1776–1786 (2003) 197. Aumi, S., Mhaskar, P.: Robust model predictive control and fault handling of batch processes. AICHE J. 57(7), 1796–1808 (2011)

References

17

198. Suárez, L.: Nonlinear MPC for fed-batch multiple phases sugar crystallization. Chem. Eng. Res. Des. 89(6), 753–767 (2011) 199. Lucia, S., Finkler, T., Engell, S.: Multi-phase nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty. J. Process Control 23(9), 1306–1319 (2013) 200. Hu, P., Yuan, P.: Robustness of state feedback predictive control systems. Control. Decis. 16, 123–127 (2001) 201. Aumi, S., Mhaskar, P.: Safe-steering of batch processes. AICHE J. 55(11), 2861–2872 (2009) 202. Wang, Y., Zhou, D., Gao, F.: Generalized predictive control of linear systems with actuator arrearage faults. J. Process Control 19(5), 803–815 (2009) 203. Hosen, M., Hussain, M., Mjalli, F.: Control of polystyrene batch reactors using neural network based model predictive control (NNMPC): an experimental investigation. Control Eng. Pract. 19(5), 454–467 (2011) 204. Wang, D.: Robust data-driven modeling approach for real-time final product quality prediction in batch process operation. IEEE Trans. Industr. Inf. 7(2), 371–377 (2011) 205. Zhang, R., Xue, A., Wang, S., et al.: An improved state-space model structure and a corresponding predictive functional control design with improved control performance. Int. J. Control 85(8), 1146–1161 (2012) 206. Zhang, R., Gao, F.: Multivariable decoupling predictive functional control with non-zero–pole cancellation and state weighting: application on chamber pressure in a coke furnace. Chem. Eng. Sci. 94(5), 30–43 (2013) 207. Aumi, S., Mhaskar, P.: An adaptive data-based modeling approach for predictive control of batch systems. Chem. Eng. Sci. 91(2), 11–21 (2013) 208. Laurí, D., Lennox, B., Camacho, J.: Model predictive control for batch processes: ensuring validity of predictions. J. Process Control, 24(1), 239–249 (2014) 209. Lee, K., Chin, I., Lee, H., et al.: Model predictive control technique combined with iterative learning for batch processes. AICHE J. 45(10), 2175–2187 (2010) 210. Shi, J., Gao, F., Wu, T.: Single-cycle and multi-cycle generalized 2D model predictive iterative learning control (2D-GPILC) schemes for batch processes. J. Process Control 17(9), 715–727 (2007) 211. Wang, Y., Dassau, E., Iii, F.: Closed-loop control of artificial pancreatic $\beta$ -cell in type 1 diabetes mellitus using model predictive iterative learning control. IEEE Trans. Biomed. Eng. 57(2), 211–219 (2010) 212. Wang, Y., Zisser, H., Dassau, E., et al.: Model predictive control with learning-type set-point: application to artificial pancreatic β-cell. AICHE J. 56(6), 1510–1518 (2010)

Chapter 2

Iterative Learning Control of Linear Batch Processes

2.1 Introduction The models of batch processes can be divided into linear systems and nonlinear systems and can also be divided into single phases and multi-phases. In this chapter, a series of research results on single-phase linear batch processes under delay and disturbances are carried out. The existence of time delays is unavoidable for both continuous and batch processes, and this often leads to unstable operation of the system. In view of this problem, a lot of research results have emerged [1–15]. However, most of these results are based on constant time delay. The time delay in the actual production process usually varies within a certain interval. There are few studies on this issue [16, 17]. Similarly, interruptions are also unavoidable in batch processes due to the fact that batch processes are typically 2D processes, whose dynamic characteristics can be divided into time and batch directions. Some perturbations only occur in the time direction and are called repetitive perturbations. Otherwise, they are called nonrepetitive perturbations. Research results on batch processes with disturbance are quite fruitful [18–22]. However, in the case of time-varying delays, the results are rare. In view of the above problems, the following issues are studied in this chapter: Firstly, for batch processes with interval time-varying delays and repeated disturbances, the state error and the output error are introduced to represent batch process as an equivalent 2D Roesser model, and the control of the system is realized by combining state feedback with ILC. Secondly, the authors find that although Roesser model and FM models are equivalent under certain conditions, FM model contains more state information and the design of the control law is more flexible. Therefore, on the basis of previous research, batch processes with non-repetitive disturbances as a 2D-FM model is further studied in this part and simulated using MATLAB. The results show that the control effect based on the 2D-FM model is better than that of the 2D Roesser model. © Springer Nature Singapore Pte Ltd. 2020 L. Wang et al., Iterative Learning Stabilization and Fault-Tolerant Control for Batch Processes, https://doi.org/10.1007/978-981-13-5790-9_2

19

20

2 Iterative Learning Control of Linear Batch Processes

Thirdly, the above two parts are implemented by state feedback combined with ILC. The premise of using this method is that the state is measurable. When the state is unmeasurable, the controller cannot be implemented. Therefore, in the end of this chapter, a control method combining output feedback with ILC is proposed, which can achieve the control of batch processes with time-varying delays and nonrepetitive disturbances when the state is unmeasurable. Finally, the injection speed and nozzle pressure in the injection molding process are simulated to verify the feasibility and effectiveness of the controller.

2.2 2D Roesser Model-Based Iterative Learning Control for Batch Processes with Uncertainties and Interval Time-Varying Delays 2.2.1 Problem Description Batch process with uncertainty and interval time-varying delay is depicted by the following equation: 

:

P−delay

⎧ x(t + 1, k)  A(t, k)x(t, k) ⎪ ⎪ ⎪ ⎨ + Ad (t, k)x(t − d(t), k) + B(t, k)u(t, k) ⎪ y(t, k)  C x(t, k) ⎪ ⎪ ⎩ x(t, k)  x0,k ; − d M ≤ t ≤ 0; k  1, 2, . . .

(2.1)

where t represents time, k expresses the cycle index, and x0,k is the initial condition of the k th batch run. x(t, k) ∈ R n , y(t, k) ∈ R l and u(t, k) ∈ R m denote, respectively, the state, output, and input at time t in the k th batch run. The time-varying delay d(t) along horizontal direction meets the following equation dm ≤ d(t) ≤ d M

(2.2)

in which dm and d M represent the lower and upper delay bounds. A(t, k)  A + a (t, k), Ad (t, k)  Ad + d (t, k), B(t, k)  B + b (t, k), {A, Ad , B, C} are constant matrices with appropriate dimensions, and {a (t, k), d (t, k), b (t, k)} are the bounds of the uncertainty that can be specified by  with

   a (t, k) d (t, k) b (t, k)  E(t, k) F Fd Fb

2.2 2D Roesser Model-Based Iterative Learning Control …

21

T (t, k)(t, k) ≤ I, 0 ≤ t ≤ T ; k  1, 2, . . . where {E, Fd , Fb } are known constant matrices, which represent the structure of the uncertainty. It is noted that (t, k) is generally expressed as the functions of both time t and cycle k. If (t, k) depends on time t only, then the uncertain parameter perturbations are called repeatable, otherwise, they are called non-repeatable.

2.2.2 Equivalent 2D System Representation For

P−delay



described above, the following ILC law is expressed:

: u(t, k)  u(t, k − 1) + r (t, k) (for u(t, 0)  0, t  0, 1, 2, . . . , T )

ilc

(2.3) where u(t, 0) is the initial profile of the iteration and r (t, k) ∈ R m is the updating law to be determined. The purpose of the ILC design is to determine the updating law r (t, k), such that y(t, k) tracks yd (t). Design δ( f (t, k))  f (t, k) − f (t, k − 1)

(2.4)

e(t, k)  y(t, k) − yd (t)

(2.5)

Based on (2.1), (2.3)–(2.5), we can be obtain 

:

2D−P−delay

  δ(x(t + 1, k)) δ(x(t, k))  A(t, k) e(t + 1, k) e(t + 1, k − 1)

 δ(x(t − d(t), k)) + B(t, k)r (t, k) (2.6) + Ad (t, k) e(t + 1, k − 1 − h(k − 1))

h m and h M indicatwhere we assume h(k − 1) satisfies h m ≤ h(k − 1) ≤ h M with  A(t, k) 0 ing the lower and upper delay bounds, A(t, k)   A + A(t, k), C A(t, k) I 



 Ad (t, k) 0 B(t, k)  Ad +Ad (t, k), B(t, k)  Ad (t, k)  B+B(t, k), C Ad (t, k) 0 C B(t, k) 



  A 0 Ad 0 B ¯ ¯ Ad (t)  , B  A  k) F, , Ad  , A(t)  E(t, C Ad 0 CA I CB

     E ¯ ¯ ¯ ¯ E(t, k) Fd , B(t)  E(t, k)Fb , E  , F¯  F 0 , and F¯d  Fd 0 . CE

22

2 Iterative Learning Control of Linear Batch Processes

The system (2.6) is essentially a special 2D Roesser system with parameter uncertainty and time-varying delays that describe the dynamic characteristics of the ILC system with convergence and tracking performance. Therefore, it is called the equivalent 2D model of the ILC system. Unlike ordinary uncertain 2D Roesser systems, the propagation of information in time direction occurs only in a limited time. However, this particularity does not prevent

2D Roesser uncertainty model-based the general robust control on 2D system 2D−P−delay . Remark 1 In order to analyze the stability of the system (2.1), we use a new vector e(t + 1, k − 1 − h(k − 1)). Then, system (2.1) can be modeled as a normal 2D system with interval time delay. Therefore, based on 2D theory, the feedback and iterative learning control (FILC) law can be introduced. Design a control law as follows:

 2D−P−delay

δ(x(t, k)) : r (t, k)  K e(t + 1, k − 1)

 (2.7)

Then the closed-loop 2D Rosser system can be obtained  2D−P−delay−C

:



 δ(x(t + 1, k)) δ(x(t, k)) ˜  A(t, k) e(t + 1, k) e(t + 1, k − 1)

 δ(x(t − d(t), k)) (2.8) + Ad (t, k) e(t + 1, k − 1 − h(k − 1))

˜ k)  A(t, k) + B(t, k)K . where A(t, The system (2.8) is expressed system (2.8) by Fig. 2.1. As we can see in Fig. 2.1, is composed of a 2D model 2D−P−delay and state feedback controller 2D−ec−delay , in which the dotted lines represent the path of information flows of the last cycle from the storages, whereas the solid lines express the path of real-time information flows. Plant P−delay is a 2D system that consists of a repetitive process and an iteration loop. On the other hand, r (t, k) is a linear function composed of δ(x(t, k)) and e(t + 1, k − 1). Both real-time information and information of the last cycle are used in the learning.   Let K  K s K l , then the following can be obtained: 

: u(t, k)  u l (t, k) + u s (t, k)

(2.9)

ILC

u l (t, k)  u(t, k − 1) + K l e(t + 1, k − 1)

(2.10)

u s (t, k)  K s δ(x(t, k))

(2.11)

The FILC law ILC consists of two parts: one is a P-type iterative learning control law where the input is adjusted proportionally with the output error, as was

2.2 2D Roesser Model-Based Iterative Learning Control …

23

Σ 2D − P − delay

Σ 2 D−C −delay u ( t , k − 1)

xe (t + 1, k − 1)

ΣC ( K )

r (t, k )

+

u (t, k )

Σ p − delay

y (t, k ) yd ( t )

x (t, k )

x ( t , k − 1)

+

e (t, k )

−

+

xe ( t , k )

Fig. 2.1 Schematic diagram of the structure of a closed-loop system (state feedback)

identified in [23], for the control improvement along cycle; the other is a real-time state feedback control law that ensures the control performance over time. According to the 2D model, a united design of these two kinds of control laws will be represented in the following description. 

 h δ(x(t + 1, k)) x (i + 1, j) , then system (2.8) is equivalent to (2.12) Let  x v (i, j + 1) e(t + 1, k)   2D−P−delay−C

:



h  

h x h (i + 1, j) ˜ k) x (i, j) + Ad (t, k) x (i − d(i), j)  A(t, x v (i, j + 1) x v (i, j) x v (i, j − h( j)) (2.12)

where the boundary conditions are as follows x h (i, j)  ρi j , ∀0 ≤ j < γ1 , −d M ≤ i ≤ 0, x v (i, j)  σi j , ∀0 ≤ i < γ2 , −h M ≤ j ≤ 0, ρ00  σ00

(2.13)

where γ1 < ∞ and γ2 < ∞ represent positive integers, ρi j and σi j are given vectors. Some research results on 2D model with delays have been studied recently such as the issues on stability or/and stabilization analysis [24–27], H∞ control [28, 29],

24

2 Iterative Learning Control of Linear Batch Processes

H∞ filtering [30, 31], guaranteed cost control [32], and so on. In the existing results, only Chen [24, 25] considers time-varying delay for which the lower bound is not restricted to be zero, and the delay-dependent stability problem for 2D systems with time-varying state saturation is studied. As far as we know, the problem of control synthesis for 2D systems with time delay has not been fully studied. It is a challenge because the stability results are difficult to generalize. The system (2.12) is a special uncertain 2D system obtained from batch process. It is different from the general 2D Rosser model because the duration of each batch is limited. The main target of this section is to design a FILC controller gain K such that the closed-loop system with interval time-varying delay can maintain stability. In order to formulate the problem, some definitions and lemmas are introduced as follows. Definition 1 [33]. For all bounded boundary conditions in (2.13), define   h  x (i, j)   χr  sup   x v (i, j)  :

 i + j  γ , i, j ≥ 1

if limr →∞ χr  0 and the system 2D−P−delay−C (2.12) is asymptotically stable, the control law (2.7) is called a 2D-control law for the system 2D−P−delay and the closed-loop system 2D−P−delay−C is called a 2D control system. Lemma 1 [34]. For any vector δ(t) ∈ R n , two positive integers κ0 , κ1 , and matrix 0 < R ∈ R n×n , the following inequality holds −(κ1 − κ0 + 1)

κ1 

δ T (t)Rδ(t) ≤ −

tκ0

κ1 

δ T (t)R

tκ0

κ1 

δ(t)

tκ0

Lemma 2 [35]. Let D, F, E, and M be real matrices of appropriate dimensions with M satisfying M  M T , then for all F T F ≤ I , M + D F E + E T F T DT < 0 if and only if there exists ε > 0 such that M + ε−1 D D T + εE T E < 0

2.2.3 Robust Stability and Control of the 2D System In this section, the main goal is to establish the delay-range dependent robust stability and robust stabilization for system (2.12) using the LMI technique. This section is mainly divided into two parts: one part is stability analysis and the other is the controller design of the 2D system (2.12).

2.2 2D Roesser Model-Based Iterative Learning Control …

2.2.3.1

25

Stability Analysis

In this section, we first analyze the stability of the system (2.12) with given K. Theorem 1 For some given scalars 0 ≤ d m ≤ d M , 0 ≤ h m ≤ h M and matrix K ∈ R m×(n+l) , the closed-loop 2D system 2D−P−delay−C (2.12) is asymptotically 

h P 0 ∈ R (n+l)×(n+l) , Q  stable if there exist symmetric positive matrices P  0 Pv

h   

h

h Q 0 M 0 G 0 (n+l)×(n+l) (n+l)×(n+l) ∈ R ∈ R ∈ , M  and G  0 Qv 0 Mv 0 Gv R (n+l)×(n+l) such that the following matrix inequality holds ⎡

⎤

1 0 G A˜ T (t, k)P Aˆ T (t, k) − I G ⎢ ⎥ T T ⎢ ⎥ 0 Ad (t, k)P Ad (t, k)G ⎢ ∗ −Q ⎥ ⎢ ⎥ 0, h¯ > 0 and matrix K ∈ R m×(n+l) , system   ¯ 2D−P−delay−C (2.25) is asymptotically stable for any time-varying delay d ∈ 0, d , 

h   P 0 ∈ R (n+l)×(n+l) , h ∈ 0, h¯ , if there exist symmetric positive matrices P  0 Pv  

h

h Q 0 G 0 (n+l)×(n+l) ∈ R ∈ R (n+l)×(n+l) such that the Q  and G  0 Qv 0 Gv following matrix inequality holds

⎤ ⎡ ¯1

G A˜ T (t, k)P A˜ T (t, k) − I G ⎢ ⎥ T ⎢ ∗ −Q − G AT (t, k)P ⎥ Ad (t, k)G d (2.26) ⎢ ⎥ 0 and h¯ > 0, a delay-dependent sufficient condition for  the existence of a FILC controller that guarantees the closed-loop 2D system 2D−P−delay−C (2.25) to be robustly stabilizable for any time-varying delay d ∈ 

h     L 0 ¯ ¯ ∈ 0, d , h ∈ 0, h is that there exist symmetric positive matrices L  0 Lv  

h

h S1 0 S2 0 ∈ R (n+l)×(n+l) , S2  ∈ R (n+l)×(n+l) , X  R (n+l)×(n+l) , S1  0 S1v 0 S2v

h  X 0 ∈ R (n+l)×(n+l) , matrix Y ∈ R m×(n+l) and positive scalars εi (i  1, 2) such 0 Xv that the following LMI holds ⎤ ⎡ ˜ 11 ˜ 12 ˜ 13 ˜ 14 ˜ 15

⎢ ∗ 0 0 ⎥ ⎥ ⎢ 22 0 ⎢ ˜ 33 0 0 ⎥ (2.29) ⎥ 0, the 2D system 2D−C−delay−C (2.33) is said to have an H∞ performance γ if it is asymptotically stable for any external disturbance ω ∈ 2D−2e , and the following condition holds z 2D−2e < γ ω 2D−2e

(2.34)

2.3.2 Robust 2D Controller Design and System Structure In this section, the main target is to design the 2D controller r (t, k) to ensure both controllability of system (2.33) and H∞ noise attenuation γ against non-repeatable perturbations at the same time. The main results are as follows. Theorem 3 Assume ω(t, k)  0, for some given scalars 0 ≤ dm ≤ d M and 0 ≤ h m ≤ h M , if there exist symmetric positive matrices L, L 1 , Q¯ 1 , Q¯ 2 , W 1 , W 2 , W˜ 1 , W˜ 2 , X 1 , X 2 ∈ R (n+l)×(n+l) , matrices Y1 , Y2 ∈ R m×(n+l) and positive scalars εi (i  1, 2, 3) such that the following LMI holds ⎡

⎤

11 12 13 ⎣ ∗ 22 0 ⎦ < 0 ∗ ∗ 33

(2.35)

where ⎡

11

⎡

12

φ11 ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ L1 0

0 0 0 0 φ12 0 0 − Q¯ 1 0 0 0 − Q¯ 2 0 0 0 0 L2 0

L1 0 0 0 φ13 0

⎤ 0 L2 ⎥ ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ ⎥ 0 ⎦ φ14

⎤ T T T T T T L 1 A1 + Y1T B L 1 A1 + Y1T B − L 1 L 1 A1 + Y1T B T T T T T T ⎢ ⎥ L 2 A2 + Y2T B L 2 A2 + Y2T B − L 2 ⎥ ⎢ L 2 A2 + Y2T B ⎢ ⎥ T T T ⎢ ⎥ L 1 Ad L 1 Ad L 1 Ad ⎢ ⎥, ⎢ ⎥ 0 0 0 ⎢ ⎥ ⎣ ⎦ 0 0 0 0 0 0

2.3 2D-FM Model-Based Iterative Learning Control for Batch …

35

⎡

22

⎤ L 1 F¯ T + Y1T FbT L 1 F¯ T + Y1T FbT L 1 F¯ T + Y1T FbT ⎢ ⎥ Y2T FbT Y2T FbT Y2T FbT ⎢ ⎥ ⎢ ⎥ L 1 F¯dT L 1 F¯dT L 1 F¯dT ⎢ ⎥

13  ⎢ ⎥, ⎢ ⎥ 0 0 0 ⎢ ⎥ ⎣ ⎦ 0 0 0 0 0 0   −2 −2 T T  diag −L + ε1 E¯ E¯ −d M X 1 + ε2 E¯ E¯ −h M X 2 + ε3 E¯ E¯ T ,  

33  diag −ε1 I −ε2 I −ε3 I , L 2  L ,

φ11  −L 1 + Q¯ 1 + (d M − dm ) Q¯ 1 + W 1 − R¯ 1 , φ12  −L + L¯ 2 + Q¯ 2 + (h M − h m ) Q¯ 2 + W 2 − R¯ 2 ,

φ13  −W˜ 1 − X 1 , φ14  −W˜ 2 − X 2 , and choose the gain of the control law that is described by (2.32) as     −1 K 2  K 11 K 12  Y1 L −1 1 Y2 L 2 then the system

2D−P−delay−C

(2.36)

(2.33) is 2D controllable.

Proof Choose the following Lyapunov function candidate V (t + θ, k + τ )  Vh (t + θ, k + τ ) + Vv (t + θ, k + τ ) where Vh (t + θ, k + τ ) 

5 

Vl (t + θ, k + τ ),

l1

Vv (t + θ, k + τ )  V1 (t + θ, k + τ ) 

10 

Vl (t + θ, k + τ )

l6 xeT (t

+ θ, k + τ )P1 xe (t + θ, k + τ ),

t+θ−1 

V2 (t + θ, k + τ ) 

xeT (r, k + τ )Q 1 xe (r, k + τ ),

r t+θ−d(t+θ)

V3 (t + θ, k + τ ) 

t+θ−1 

xeT (r, k + τ )W1 xe (r, k + τ ),

r t+θ−d M

V4 (t + θ, k + τ ) 

−dm 

t+θ−1 

xeT (r, k + τ )Q 1 xe (r, k + τ ),

s−d M r t+θ+s

V5 (t + θ, k + τ )  d M

−1 

t+θ−1 

s−d M r t+θ+s

η1T (r, k + τ )R1 η1 (r, k + τ ),

(2.37)

36

2 Iterative Learning Control of Linear Batch Processes

Vv (t + θ, k + τ )  V6 (t + θ, k + τ ) 

10 

Vl (t + θ, k + τ ),

l6 xeT (t

+ θ, k + τ )(P − P1 )xe (t + θ, k + τ ),

k+τ −1 

V7 (t + θ, k + τ ) 

xeT (t + θ, j)Q 2 xe (t + θ, j),

jk+τ −h(k+τ ) k+τ −1 

V8 (t + θ, k + τ ) 

xeT (t + θ, j)W2 xe (t + θ, j),

jk+τ −h M −h m 

V9 (t + θ, k + τ ) 

k+τ −1 

xeT (t + θ, j)Q 2 xe (t + θ, j),

s−h M jk+τ +s

V10 (t + θ, k + τ )  h M

−1 

k+τ −1 

η1T (t + θ, j)R2 η1 (t + θ, j),

s−h M jk+τ +s

η1 (r, k)  xe (r + 1, k) − xe (r, k), η1 (t + 1, j)  xe (t + 1, j + 1) − xe (t + 1, j), and P1 , P, Q 1 , Q 2 , W1 , W2 , R1 and R2 are positive definite matrices to be determined. Design V (t + 1, k)  Vh (t + 1, k) − Vh (t, k) + Vv (t + 1, k) − Vv (t + 1, k − 1) 

10 

Vl (t + 1, k)

(2.38)

l1

which yields V1 (t + 1, k)  xeT (t + 1, k)P1 xe (t + 1, k) − xeT (t, k)P1 xe (t, k)

(2.39a)

V2 (t + 1, k) ≤ xeT (t, k)Q 1 xe (t, k) − xeT (t − d(t), k)Q 1 xe (t − d(t), k) +

t−dm

Σ

r t−d M

xeT (r, k)Q 1 xe (r, k) (2.39b)

V3 (t + 1, k)  xeT (t, k)W1 xe (t, k) − xeT (t − d M , k)W1 xe (t − d M , k) V4 (t + 1, k)  (d M − dm )xeT (t, k)Q 1 xe (t, k) −

t−d m

(2.39c)

xeT (r, k)Q 1 xe (r, k) (2.39d)

r t−d M 2 T V5 (t + 1, k)  d M η1 (t, k)R1 η1 (t, k) − d M

t−1  r t−d M

η1T (r, k)R1 η1 (r, k)

(2.39e)

2.3 2D-FM Model-Based Iterative Learning Control for Batch …

37

V6 (t + 1, k)  xeT (t + 1, k)P2 xe (t + 1, k) − xeT (t + 1, k − 1)P2 xe (t + 1, k − 1) (2.39f) V7 (x1 (t, k)) ≤ xeT (t + 1, k − 1)Q 2 xe (t + 1, k − 1) − xeT (t + 1, k − 1 − h(k − 1))Q 2 xe (t + 1, k − 1 − h(k − 1)) +

k−1−h m

xeT (t + 1, j)Q 2 xe (t + 1, j)

(2.39g)

jk−1−h M

V8 (t + 1, k) ≤ xeT (t + 1, k − 1)W2 xe (t + 1, k − 1) − xeT (t + 1, k − 1 − h M )W2 xe (t + 1, k − 1 − h M )

(2.39h)

V9 (t + 1, k)  (h M − h m )xeT (t + 1, k − 1)Q 2 xe (t + 1, k − 1) −

k−1−h m

xeT (t + 1, j)Q 2 xe (t + 1, j)

(2.39i)

jk−1−h M V10 (t + 1, k)  h 2M η1T (t + 1, k − 1)R2 η1 (t + 1, k − 1) k−2  η1T (t + 1, j)R2 η1 (t + 1, j) − hM jk−1−h M

(2.39j)

Using Lemma 1, we have t−1 

2 T V5 (t + 1, k) ≤ d M η1 (t, k)R1 η1 (t, k) −

η1T (r, k)R1

r t−d M

t−1 

η1 (r, k)

r t−d M

2  dM (xe (t + 1, k) − xe (t, k))T R1 (xe (t + 1, k) − xe (t, k))

− (xe (t, k) − xe (t − d M , k))T R1 (xe (t, k) − xe (t − d M , k)) (2.40a) and V10 (t + 1, k) ≤ h 2M η1T (t + 1, k − 1)R2 η1 (t + 1, k − 1) −

k−2

Σ

jk−1−d M

η1T (t + 1, j)R2

k−2

Σ

jk−1−d M

η1 (t + 1, j)

 h 2M (xe (t + 1, k) − xe (t + 1, k − 1))T R2 (xe (t + 1, k) − xe (t + 1, k − 1)) − (xe (t + 1, k − 1) − xe (t + 1, k − 1 − h M ))T R2 (xe (t + 1, k − 1) − xe (t + 1, k − 1 − h M )) (2.40b)

It follows from (2.38)–(2.39j) that V (t + 1, k) ≤ ϕ T (t, k) ϕ(t, k)

(2.41)

2

 1 + T1 P 1 + d M T2 R1 2 + h 2M T3 R2 3

(2.42)

38

2 Iterative Learning Control of Linear Batch Processes

where     ϕ T (t, k)  ϕ1T (t, k) ϕ2T (t, k) ϕ3T (t, k) , ϕ1T (t, k)  xeT (t, k) xeT (t + 1, k − 1) ,   ϕ2T (t, k)  xeT (t − d(t), k) xeT (t + 1, k − 1 − h(k − 1)) ,   ϕ3T (t, k)  xeT (t − d M , k) xeT (t + 1, k − 1 − h M ) , ⎤ −P1 + Q 1 + ⎥ ⎢ (d M − dm )Q 1 + 0 0 0 R1 0 ⎥ ⎢ ⎥ ⎢ W1 − R 1 ⎥ ⎢ ⎥ ⎢ −P + P1 + Q 2 + ⎥ ⎢ ⎥ ⎢ 0 0 R2 0 (h M − h m )Q 2 + 0 ⎥ ⎢

1  ⎢ ⎥ ⎥ ⎢ W2 − R 2 ⎥ ⎢ ⎥ ⎢ 0 0 −Q 0 0 0 1 ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 −Q 2 ⎥ ⎢ ⎦ ⎣ 0 0 0 −W1 − R1 0 R1 0 0 0 −W2 − R2 0 R2   1  A˜ 1 (t, k) A˜ 2 (t, k) Ad (t, k) 0 0 0 ,   2  ( A˜ 1 (t, k) − I ) A˜ 2 (t, k) Ad (t, k) 0 0 0 ,   3  A˜ 1 (t, k) ( A˜ 2 (t, k) − I ) Ad (t, k) 0 0 0 , P2  P − P1 > 0. ⎡

In order to prove < 0, first, pre- and post-multiply (2.35) by diag[P1 , P2 , P1 , P2 , R1 , R2 , P, R1 , R2 , I, I, I ] and define L  P −1 , L 1  P1−1 , L 2  L, X 1  R1−1 , X 2  R2−1 , L 1 W1 L 1  W 1 , L 2 W2 L 2  W 2 ,   L 1 Q 1 L 1  Q¯ 1 , L 2 Q 2 L 2  Q¯ 2 , X 1 W1 X 1  W˜ 1 , X 2 W2 X 2  W˜ 2 , Y  Y1 Y2    K 1 L 1 K 2 L 2 , and let R¯ i and L¯ 2 represent L i Ri L i (i  1, 2) and L 2 P1 L 2 , respectively. Then, using Schur complements and Lemma 2, the inequality (2.35) is trans⎤ ⎡

11 12 13 14 ⎢ ∗ 22 23 24 ⎥ ⎥ formed into inequality (2.42), that is, ⎢ ⎣ ∗ ∗ 33 0 ⎦ < 0 is equivalent to

< 0. Hence, V (t + 1, k) < 0 . For any ϕ(t, k)  0, we have:

∗

∗

∗ 44

Vh (t + 1, k) + Vv (t + 1, k) ≤ Vh (t, k) + Vv (t + 1, k − 1) with Vh (t, k) + Vv (t + 1, k − 1)  xeT (t, k)P1 xe (t, k) + xeT (t + 1, k − 1)P2 xe (t + 1, k − 1)

(2.43)

2.3 2D-FM Model-Based Iterative Learning Control for Batch …

+ +

t−1

xeT (r, k)Q 1 xe (r, k) +

Σ

r t−d(t) t−1

Σ

r t−d M −dm

xeT (r, k)W1 xe (r, k) +

k−2

Σ

jk−1−h M

xeT (t + 1, j)W2 xe (t + 1, j)

−h m

t−1

k−2

Σ xeT (r, k)Q 1 xe (r, k) + Σ

+ Σ

s−d M r t+s −1

Σ

s−h M jk−1+s

xeT (t + 1, j)Q 2 xe (t + 1, j)

−1

t−1

Σ η1T (r, k)R1 η1 (r, k) + h M

+ dM Σ

xeT (t + 1, j)Q 2 xe (t + 1, j)

Σ

jk−1−h(k−1) k−2

39

k−2

Σ

s−d M r t+s

Σ

s−h M jk−1+s

η1T (t + 1, j)R2 η1 (t + 1, j)

When the boundary condition is met in (2.13), for any nonnegative integer N, we have 

V (t, k)

t+kN +1

 V (N , 1) + V (N − 1, 2) + · · · + V (2, N − 1) + V (1, N ) ≤ xeT (N − 1, 1)P1 xe (N − 1, 1) + xeT (N , 0)P2 xe (N , 0) + +

N −1−1

−1

Σ

r N −1−d(N −1) N −1−1

j−h M −1

Σ

r N −1−d(N −1) −dm

xeT (r, 1)Q 1 xe (r, 1) + Σ

xeT (r, 1)W1 xe (r, 1) + Σ

N −1−1

+ Σ

Σ

s−d M r N −1+s −1

+ dM Σ

j−h M

N −1−1

Σ

s−d M r N −1+s

+ hM

−1

Σ

−h m

xeT (r, 1)Q 1 xe (r, 1) + Σ

xeT (N , j)Q 2 xe (N , j) xeT (N , j)W2 xe (N , j) −1

Σ xeT (N , j)Q 2 xe (N , j)

s−h M j+s

η1T (r, 1)R1 η1 (r, 1)

−1

Σ η1T (N , j)R2 η1 (N , j)

s−h M j+s

+ · · · + xeT (0, N )P1 xe (0, N ) + xeT (1, N − 1)P2 xe (1, N − 1) +

−1

Σ

r −d(t)

xeT (r, N )Q 1 xe (r, N ) +

N −2

Σ

jN −1−h(N −1)

−1

N −2

r −d M

jN −1−h M

+ Σ xeT (r, N )W1 xe (r, N ) + −dm

+ Σ

Σ

−h m

−1

Σ xeT (r, N )Q 1 xe (r, N ) + Σ

s−d M r +s −1

+ dM Σ

xeT (1, j)Q 2 xe (1, j)

xeT (1, j)W2 xe (1, j) N −2

Σ

s−h M jN −1+s

xeT (1, j)Q 2 xe (1, j)

−1

Σ η1T (r, N )R1 η1 (r, N )

s−d M r s

+ hM β

−1

Σ

N −2

Σ

s−h M jN −1+s

η1T (1, j)R2 η1 (1, j) (2.44)

40

2 Iterative Learning Control of Linear Batch Processes

According to (2.37), let θ  τ  0 and k returns to k − 1, then the Lyapunov functional becomes V (t, k − 1)  Vh (t, k − 1) + Vv (t, k − 1)  xeT (t, k − 1)P xe (t, k − 1) t−1 

+

k−2 

xeT (r, k − 1)Q 1 xe (r, k − 1) +

r t−d(t) t−1 

+

jk−1−h(k−1)

+

k−2 

xeT (r, k − 1)W1 xe (r, k − 1) +

r t−d M −dm 

xeT (t, j)Q 2 xe (t, j) xeT (t, j)W2 xe (t, j)

jk−1−h M t−1 

xeT (r, k − 1)Q 1 xe (r, k − 1)

s−d M r t+s −h m 

+

k−2 

xeT (t, j)Q 2 xe (t, j)

s−h M jk−1+s t−1 −1  

+ dM

η1T (r, k − 1)R1 η1 (r, k − 1)

s−d M r t+s −1 

+ hM

k−2 

η1T (t, j)R2 η1 (t, j)

(2.45)

s−h M jk−1+s

Thus β  V (N , 0) + V (N − 1, 1) + · · · + V (1, N − 1) + V (0, N ) − xeT (N , 0)P1 xe (N , 0) − xeT (0, N )P2 xe (0, N ) −

N −1 

xeT (r, 0)Q 1 xe (r, 0) −

r N −d(N )

−

N −1 

−

N −1 

xeT (r, 0)W1 xe (r, 0) −

xeT (0, j)Q 2 xe (0, j)

xeT (0, j)W2 xe (0, j)

jN −h M N −1 

xeT (r, 0)Q 1 xe (r, 0)

s−d M r N +s

− dM

N −1 

j−h M jN −h(N )

r N −d M −dm 

−1 

N −1 −1  

−

−h m 

N −1 

xeT (0, j)Q 2 xe (0, j)

s−h M jN +s

η1T (r, 0)R1 η1 (r, 0) − h M

s−d M r N +s

−1 

N −1 

η1T (0, j)R2 η1 (0, j)

s−h M jN +s

≤ V (N , 0) + V (N − 1, 1) + · · · + V (1, N − 1) + V (0, N ) From (2.44) and (2.46), we have

(2.46)

2.3 2D-FM Model-Based Iterative Learning Control for Batch …



V (t, k) ≤

t+kN +1



V (t, k)

41

(2.47)

t+kN

Then, from [33], one can conclude that lim xe (t, k) → 0

t+K →∞

(2.48)

which indicates that system (2.33) is asymptotically stable. This completes the proof of Theorem 3. Theorem 4 For some given scalars 0 ≤ dm ≤ d M , 0 ≤ h m ≤ h M , and γ > 0, if there exist symmetric positive matrices L, L 1 , Q¯ 1 , Q¯ 2 , W 1 , W 2 , W˜ 1 , W˜ 2 , X 1 , X 2 ∈ R (n+l)×(n+l) , matrices Y1 , Y2 ∈ R m×(n+l) and positive scalars εi (i  1, 2, 3) such that the following LMI holds ⎡ ˜

11 ⎢ ∗ ⎢ ⎣ ∗ ∗

˜ 12

22 ∗ ∗

˜ 13

0

33 ∗

˜ 14 ⎤

0 ⎥ ⎥ 0 and h¯ > 0, a delay-dependent sufficient  the  condition for the existence of a 2D controller (2.32) that guarantees system 2D−P−delay−C (2.56) to be robustly stabilizable for any delays d ∈ 0, d¯ ,   h ∈ 0, h¯ is that there exist symmetric positive matrices L, L 1 , Q¯ 1 , Q¯ 2 , Q˜ 1 , Q˜ 2 , X 1 , X 2 ∈ R (n+l)×(n+l) , matrices Y1 , Y2 ∈ R m×(n+l) and positive scalars εi (i  1, 2, 3) such that the following LMI holds

44

2 Iterative Learning Control of Linear Batch Processes

⎡

⎤ ˆ 11 ˆ 12 ˆ 13

⎣ ∗ ˆ 22 0 ⎦ < 0 ∗ ∗ 33

(2.57)

where

ˆ 11

⎡ˆ φ11 ⎢ 0 ⎢ ⎣ L1 ⎡

ˆ 12

ˆ 13

ˆ 22

φˆ 11

0 ˆ φ12 0 0 L2

L1 0 φˆ 13 0

⎤ 0 L2 ⎥ ⎥, 0 ⎦ φˆ 14

⎤ T T T T T T L 1 A1 + Y1T B L 1 A1 + Y1T B − L 1 L 1 A1 + Y1T B T T T T T T ⎢ ⎥ ⎢ L A + YTB L 2 A2 + Y2T B L 2 A2 + Y2T B − L 2 ⎥  ⎢ 2 2 T2 ⎥, T T ⎣ ⎦ X 1 Ad X 1 Ad X 1 Ad 0 0 0 ⎤ ⎡ T T T T T T T T T L 1 F¯ + Y1 Fb L 1 F¯ + Y1 Fb L 1 F¯ + Y1 Fb ⎥ ⎢ Y2T FbT Y2T FbT Y2T FbT ⎥, ⎢ T T T ⎦ ⎣ ¯ ¯ ¯ X 1 Fd X 1 Fd X 1 Fd 0 0 0    diag −L + ε1 E¯ E¯ T −d¯ −2 X 1 + ε2 E¯ E¯ T −h¯ −2 X 2 + ε3 E¯ E¯ T ,  −L 1 + Q¯ 1 − R¯ 1 ,

φˆ 12  −L + L¯ 2 + Q¯ 2 − R¯ 2 , φˆ 13  − Q˜ 1 − X 1 , φˆ 14  − Q˜ 2 − X 2 . Corollary 4 Given scalars d¯ > 0 and h¯ > 0, system 2D−P−delay−C (2.56) is 2D controllable via a 2D control law(2.32)  and has an H∞ performance less than γ for any delays d ∈ 0, d¯ and h ∈ 0, h¯ if there exist symmetric positive matrices L, L 1 , Q¯ 1 , Q¯ 2 , Q˜ 1 , Q˜ 2 , X 1 , X 2 ∈ R (n+l)×(n+l) , matrices Y1 , Y2 ∈ R m×(n+l) and positive scalars εi (i  1, 2, 3) such that the following LMI holds ⎡

11 ⎢ ∗ ⎢ ⎣ ∗

12

13

ˆ 22 0

∗ 33 ∗ ∗ ∗

14 ⎤

0 ⎥ ⎥ 0, Q i > 0, W i > 0, Wi > 0, X i > 0, ˜ ˜ ≥ 0, Li Xi     

 



¯ L 2 L˜ 2 ≥ 0, R¯ i I ≥ 0, L i I ≥ 0, L 2 I ≥ 0, X i I ≥ 0,  I R˜ i I L˜ i I X˜ i L˜ 2 L˜ 1 I L2 εi >0 (i  1, 2).

(2.59)

Using the linearization method proposed in [37], a suboptimal maximal delay can be obtained by an iterative algorithm. If the above restriction has a feasible be proposed again as far as a solution, a larger delay boundary d M and h M can satisfactory solution appears. Then the control law 2D−ec−delay−F can be given by   −1 . Y1 L −1 1 Y2 L 2 Case 2: ω(t, k)  0. Under this situation, the above design goals should be sustained and Theorem 4 should be used to design the controller 2D−ec−delay−F with H∞ performance less than γ . An iterative algorithm is used to find a suboptimal maximal delay for some prescribed γ or a suboptimal γ for some prescribed delay to achieve good H∞ performance of the system. A more practical method for the solution is to assume that acceptable lower and upper delay bounds are prescribed, resulting in the following robust H∞ performance optimization problem:

46

2 Iterative Learning Control of Linear Batch Processes

Minimize γ 2 .



R˜ i L˜ i ¯ ˜ Subject to (2.49) and L i > 0, Q i > 0, W i > 0, Wi > 0, X i > 0, ˜ ˜ ≥ 0, Li Xi     

 



¯ L 2 L˜ 2 ≥ 0, R¯ i I ≥ 0, L i I ≥ 0, L 2 I ≥ 0, X i I ≥ 0,  I R˜ i I L˜ i I X˜ i L˜ 2 L˜ 1 I L2 εi >0 (i  1, 2).

(2.60)

2.4 Delay-Range-Dependent Robust 2D Output Feedback Iterative Learning Control for Batch Processes with State Delay and Uncertainties Based on 2D-FM Model The research in Sects. 2.2 and 2.3 are based on state feedback, while this method can only be used when the state is measurable. When the state is unmeasurable, this method is no longer applicable. Therefore, in this section, a control method combining output feedback with traditional ILC is proposed, which overcomes the shortcomings of state feedback, achieves the desired control objectives and achieves the corresponding control performance.

2.4.1 Equivalent 2D System Representation Utilizing the method similar to Sect. 2.3, the augmented model can be derived as  : 2D−P−delay

  ⎧ k) + A2 xe (t + 1, k − 1) ⎪ ⎪ x1e (t + 1, k)  A1 + A(t, k) xe (t, ⎪    ⎪ ⎪ ⎪ + Ad + Ad (t, k) xe (t − d(t), k) + B + B(t, k) r (t, k) ⎪ ⎪ ⎪ ⎨ + H¯ ω(t, k)

 ⎪ ⎪ e(t, k − 1) ⎪ ⎪ y(t, k)   C xe (t, k) ⎪ ⎪ e(t, k) ⎪ ⎪ ⎪ ⎩ ¯ e (t, k) z(t, k)  e(t, k)  Gx (2.61) Note that (2.61) is different from (2.33) as there is no xe (t + 1, k − 1 − h(k − 1)) in it and it is not the strictly 2D-FM model but just similar to a 2D-FM model. 2D−P−delay is called an equivalent 2D tracking error model for system (2.1), because this model equivalently expresses the dynamic behavior of the tracking error in (2.1).

2.4 Delay-Range-Dependent Robust 2D Output Feedback Iterative Learning …

47

2.4.2 Robust Stability and Control of a 2D-FM System The main goal in the current section is to establish the delay-range-dependent robust stability and output feedback control for system (2.61) using the LMI technique. Two issues are proposed: the stability analysis and the output feedback controller design of the 2D system (2.61).

2.4.2.1

Robust Stability and H∞ Performance Analysis

In this section, the stability and H∞ performance analysis for the 2D system (2.61) are discussed. A sufficient delay-range-dependent condition for the asymptotic stability of the unforced 2D system (2.61) is proposed for repetitive perturbation, that is, ω(t, k)  0. Afterwards, according to non-repetitive perturbation, i.e., ω(t, k)  0, the H∞ performance of the system is considered. Theorem 5 The 2D system (2.61) is robustly asymptotically stable for any delay satisfying 0 ≤ dm ≤ d(t) ≤ d M , if symmetric positive matrices P, P1 , Q 1 , R1 and S1 ∈ R (n+l)×(n+l) and positive scalars εi (i  1, 2) exist such that the following LMI holds ⎡

1 ⎢ ∗ ⎢ ⎣ ∗ ∗ where

T11 T22 ∗ ∗

T12 0 T33 ∗

⎤ 0 T23 ⎥ ⎥ 0, Q˜ 1 > 0, R˜ 1 > 0, Dci , Z i , Z¯ i , Z˜ 1 , and Zˆ i (i  1, 2) and positive scalars εk (k  1, 2) such that the following LMI holds ⎡ ¯

1 ⎢ ∗ ⎢ ⎣ ∗ ∗

T¯ 11 T¯ 12 T¯ 22 0 ∗ T33 ∗ ∗

⎤ 0 T¯ 23 ⎥ ⎥ 0, Q˜ 1 > 0, R˜ 1 > 0, Dci , Z i , Z¯ i , Z˜ 1 , Zˆ i (i  1, 2) and positive scalars εk (k  1, 2) exist such that the following LMI holds ⎡ ⎤ ¯ 1 0 T¯ 11 T¯ 12 0 T¯ 15

⎢ ∗ −γ 2 I T¯ 0 0 ⎥ 21 0 ⎢ ⎥ ⎢ ⎥ ¯ ¯ ∗ T22 0 T23 0 ⎥ ⎢ ∗ (2.69) ⎢ ⎥ 0, Y > 0, Q˜ 1 > 0, Dci , Z i , Z¯ i , Z˜ 1 , Zˆ i (i  1, 2) and positive scalars εk (k  1, 2) exist such that the following LMI holds:

54

2 Iterative Learning Control of Linear Batch Processes

⎡ ˜ ˜

1 T11 ⎢ ∗ T˜ 22 ⎢ ⎣ ∗ ∗ ∗ ∗

T˜ 12 0 T33 ∗

⎤ 0 T¯ 23 ⎥ ⎥ 0, Y > 0, Q˜ 1 > 0, Dci , Z i , Z¯ i , Z˜ 1 , Zˆ i (i  1, 2) and positive scalars εk (k  1, 2) exist such that the following LMI holds: ⎡ ⎤ ˜ 1 0 T˜ 11 T˜ 12 0 T˜ 15

⎢ ∗ −γ 2 I T¯ 0 0 ⎥ 21 0 ⎢ ⎥ ⎢ ⎥ ∗ T˜ 22 0 T¯ 23 0 ⎥ ⎢ ∗ (2.73) ⎢ ⎥ 0 , i  1, 2, . . . , r , we have i1 h i (z(t, k))  1 , i  wi (z(t, k)) ≥ 0 h i (z(t, k)) ≥ 0 1, 2, . . . , r . Remark 11 The 1D local sector nonlinearity method proposed in [20] can construct the T-S fuzzy model in each batch, so it can be applied to the 2D batch process. For the nonlinear model with no uncertainty, the 2D T-S fuzzy model, as shown in (3.3), can be obtained by extending the fuzzy model of each batch to all the batches. Unlike the 1D case, the running time of the 2D batch process is limited, the system dynamics evolutes along two directions, which make the model become more complex as well as the analysis and synthesis of controller design. For the 2D T-S fuzzy model represented by Eq. (3.3), the control objective is to find an input u(t, k) such that the output of the system tracks the set point or the expected profile.

3.2.2 Robust Fuzzy Iterative Learning Controller Design Exploiting the repeatability of batch processes, ILC can be used as a base-line controller in batch direction. Considering the time-varying uncertainty from one cycle to another and the non-repetitive load disturbance, a feedback control scheme is introduced in the time direction. Section 3.2.2.1 is mainly about the description of 2D T-S fuzzy systems and the 2D FILC law is designed in Sect. 3.2.2.2. The control system structure is given in Sect. 3.2.2.3, and the control algorithm is in Sect. 3.2.2.4. In the end, two cases are given in Sect. 3.4.1 to show the efficiency of the controller.

3.2.2.1

An Equivalent 2D T-S Fuzzy System Description

When an ILC strategy is adopted, the ILC law should have the general form as follows:

68

3 Iterative Learning Control of Nonlinear Batch Processes

  u(t, k)  u(t, k − 1) + r (t, k) : u(t, 0)  0

(3.4)

f −ilc

Define the system state error in the adjacent cycles as (3.5) and the output tracking error as (3.6): δ(x(t, k))  x(t, k) − x(t, k − 1)

(3.5)

e(t + 1, k)  yr (t + 1, k) − y(t + 1, k)

(3.6)

Based on (3.4)–(3.6), we have δ(x(t + 1, k))  x(t + 1, k) − x(t + 1, k − 1) 

r 

h i (z(t, k))Ai x(t, k) +

i1 r 

− − 

i1 r 

h i (z(t, k))Bi u(t, k) + ω(t, k)

i1

h i (z(t, k − 1))Ai x(t, k − 1) h i (z(t, k − 1))Bi u(t, k − 1) − ω(t, k − 1)

i1 r 

h i (z(t, k))Ai (x(t, k) − x(t, k − 1))

i1 r 

+

+ − − 

r 

i1 r 

h i (z(t, k))Bi (u(t, k) − u(t, k − 1)) h i (z(t, k))Ai x(t, k − 1) +

i1 r  i1 r 

r 

h i (z(t, k))Bi u(t, k − 1)

i1

h i (z(t, k − 1))Ai x(t, k − 1) h i (z(t, k − 1))Bi u(t, k − 1) + δ(ω(t, k))

i1 r 

r 

i1

i1

h i Ai δ(x(t, k)) +

h i Bi r (t, k) + w(t, k) + δ(ω(t, k)) (3.7)

where w(t, k) 

r  i1

h i (z(t, k))Ai x(t, k − 1) +

r  i1

h i (z(t, k))Bi u(t, k − 1)

3.2 2D T-S Fuzzy Model-Based Iterative Learning Control …

− 

r 

h i (z(t, k − 1))Ai x(t, k − 1) +

i1 r 

r 

i1

i1

r 

69

h i (z(t, k − 1))Bi u(t, k − 1)

i1

δ(h i )Ai x(t, k − 1) +

δ(h i )Bi u(t, k − 1)

For simplicity, let h i stand for h i (z(t, k)), and δ(h i ) is expressed as δ(h i )  h i (z(t, k)) − h i (z(t, k − 1)). For the output tracking error, a special condition is considered, Ci  C, i  1, 2, . . . , r . Then, from (3.4), (3.5), and (3.7), the following is derived e(t + 1, k)  yr (t + 1) − y(t + 1, k)  e(t + 1, k − 1) − C(x(t + 1, k) − x(t + 1, k − 1))  e(t + 1, k − 1) r

r   −C h i Ai (x(t, k)) + h i Bi r (t, k) + w(t, k) + δ(ω(t, k)) i1

i1

(3.8) Furthermore, the equivalent 2D system description can be expressed as    r ⎧ r   δ(x(t + 1, k)) δ(x(t, k)) ⎪ ⎪  h A h i B i r (t, k) + Dw(t, k) + i i  ⎨ e(t + 1, k) e(t + 1, k − 1) i1 i1   : ⎪ δ(x(t, k)) ∧ 2D−T-S ⎪ ⎩ z(t, k)  e(t + 1, k − 1)  C e(t + 1, k − 1) (3.9)       Ai 0 Bi I , D , Bi  Ai  , C  [0 1]; −C Bi −C Ai I −C w(t, k)  (w(t, k) + δ(ω(t, k))).

3.2.2.2

2D FILC Law Design

The updating law r (t, k) is designed by parallel distributed compensation (PDC) method. The expression form is as follows: Rule i:  r (t, k)  K i

δ(x(t, k)) e(t + 1, k − 1)

 (3.10)

70

3 Iterative Learning Control of Nonlinear Batch Processes

Then, the whole 2D T-S fuzzy updating law is obtained: 

: r (t, k) 

2D−T-S−ilc

r 

 hi Ki

i1

δ(x(t, k)) e(t + 1, k − 1)

 (3.11)

   δ(x(t + 1, k)) xh (t + 1, k)  , then based on (3.9) and (3.11), the 2D xv (t, k + 1) e(t + 1, k) T-S fuzzy closed-loop system is as follows:      r  r

 xh (t, k) xh (t + 1, k)  h i h j Ai + B i K j + Dw(t, k) xv (t, k + 1) xv (t, k) i1 j1   r 

 xh (t, k) 2  h i Ai + B i K i + Dw(t, k) xv (t, k) i1

  r   xh (t, k) Ai + B i K j A j + B j K i hi h j +2 + (3.12) 2 2 xv (t, k) i1 i< j 

Let

Let G icj  Ai + B i K j , (3.12) can be rewritten as:     r  xh (t, k) xh (t + 1, k) 2 c h i G ii  xv (t, k + 1) xv (t, k) i1   c  r   G i j + G cji xh (t, k) hi h j +2 + Dw(t, k) 2 xv (t, k) i1 i< j

Let x t,k



 

(3.13)

   xh (t + 1, k) xh (t, k) , x t,k  x(t, k)  , system (3.13) can be xv (t, k + 1) xv (t, k)

rewritten as: 

:

2D−T−S−C



x t,k





r  i1

h i2 G iic x t,k + 2

r   i1 i< j

 hi

G icj + G cji 2

 x t,k + Dw(t, k)

(3.14)

According to the description above, since the system dynamics change along batch direction and the time direction, which is defined as T direction and K direction, respectively, the boundary condition also needs to be a two-dimensional boundary condition T

x 0  x h (0, k)T x v (t, 0)T , t, k  1, 2, . . .

(3.15)

3.2 2D T-S Fuzzy Model-Based Iterative Learning Control …

71

Here x h (0, k) and x v (t, 0) represent the T -boundary and K-boundary, respectively. Under design the updating  any 2D boundary conditions, the first objective is to law 2D−T−S−ilc , in order to make sure the closed-loop system 2D−T−S−C satisfies the following conditions: (1) 2D robust stabilizable along both T and K directions with convergence index; (2) Least possible 2D robust H∞ performance γ under nonzero disturbance.  Definition 3 2D T-S fuzzy closed-loop system 2D−T−S−C  with w(t, k)  0 is x t,k   0 for any boundcalled 2D stabilizable if the system state satisfies lim t,k→∞  ary condition (3.15), system 2D−T−S is called 2D stabilizable system, and FILC  law 2D−T−S−ilc is called the 2D T-S fuzzy control law. Definition 3 gives the conditions for the overall stabilization of the system. The following definition is given to describe disturbance rejection ability of the system  2D−T−S−C under disturbance w(t, k) quantitatively.  Definition 4 The 2D T-S fuzzy closed-loop system 2D−T-S−delay−C is fuzzy H∞ control, if the following conditions hold and there exists a scalar γ > 0 for any w(t, k) (1) the resulting closed-loop system with w(t, k)  0 is asymptotically stable; (2) with the zero boundary conditions, and for any disturbance w ∈ l2D−T-S−delay−C−2e , if the controlled output satisfies,

z 2D−T-S−delay−C−2e < γ w 2D−T-S−delay−C−2e At the same time, the system performance γ .

 2D−T-S−delay−C

(3.16)

is called to have robust H∞

Remark 12 The robust H∞ stability γ represents the sensitivity of the output to the external disturbance of the system. The smaller the γ , the better the performance is. Thus, γ should be optimized to be minimum when the controller is designed. The Lyapunov function method is an effective method for stability and convergence analysis of control systems. This chapter will also use the 2D Lyapunov function method to analyze the stability of the closed-loop system 2D−T-S−C . First, we need to extend corollary 3 and 4 in [40] and Lemma 3 is given as follows: Lemma 3 [40] If the number of the fuzzy rules for all t is less than or equal to s(1 < s ≤ r ), then the following inequality holds r  i1

where

r  i1

1  2h i h j ≥ 0 s − 1 i1 i< j r

h i2 −

h i  1, h i ≥ 0 for all i.

(3.17)

72

3 Iterative Learning Control of Nonlinear Batch Processes

Then, the stability analysis and the convergence index of the system under nonzero interference are given below.  Lemma 4 2D T-S fuzzy closed-loop system 2D−T-S−C is 2D stabilizable with w(t, k)  0, if there exists Lyapunov function V (·) such that: (a) V (x(t, k)) ≥ 0 for ∀x(t, k), and V (x(t, k))  0 ⇔ x(t, k)  0; (b) V (x(t, k)) → ∞, as x(t, k) → ∞; (c) For any boundary conditions, if there exists a scalar 0 < ρ < 1 such that 

V (x(t, k)) < ρ

t + k  T0 + K 0 + m + 1 T0 ≤ t ≤ T0 + k K0 ≤ k ≤ K0 + k



V (x(t, k)),

t + k  T0 + K 0 + m T0 ≤ t ≤ T0 + m K0 ≤ k ≤ K0 + m

∀T0 > 0, K 0 > 0, m > 0

(3.18)

the minimum of ρ is called 2D convergence index of the system. Remark 13 α and β are T and K convergence indexes, respectively, which quantify the stability along T and K direction. Better convergence performance along T and K means the smaller α and β. The introduction of these two convergence indexes provides quantitative basis for evaluating the control performance in different directions. At the same time, these two indexes reflect the convergence speed in batch and time directions and can be optimized in different directions to meet the required control performance. In Lemma 4, ρ  max{α, β}. Theorem 9 For given scales 0 < α < 1, 0 < β 0, if there exist symmetric positive definite matrices i , matrices Y1i , Y2i ∈ Rm×(n+2l) , and scalars μi ≥ 1, εai , εbi > 0 such that the following LMI holds, respectively iT ⎤ iT iT −ηt1 i 0 i AiT i E 1iT Y1iT E b 1 + Y1 Bb iT iT ⎥ ⎢ −ηt2 i i A2 + Y2iT BbiT 0 Y2iT E b ⎥ ⎢ ∗ ⎥ ⎢ i iT 0,  t2 > 0, and when wi (t, k)  0, the following will be drawn according to (4.27a): V i (x(t, k)) ≤ Vhi (x h (t + 1, k)) + Vvi (xv (t + 1, k)) − ηi Vhi (x h (t, k)) − ηi Vvi (xv (t + 1, k − 1))  x hiT (t + 1, k)t1 Ph x hi (t + 1, k) + xviT (t + 1, k)t2 Pv (xvi (t + 1, k) − ηi x hiT (t, k)t1 Ph x hi (t, k) − ηi xviT (t + 1, k − 1)t2 Ph (xvi (t + 1, k − 1)) ⎞ ⎛⎡  i T ⎤  T     A  x t,k x t,k ⎝⎣  1n T ⎦ P i Ai Ai − ηi diag t1 Phi , t2 Pvi ⎠  1n 2n i x t+1,k−1 x t+1,k−1 A 2n

 

x t,k x t+1,k−1

T

 i

x t,k



x t+1,k−1

(4.34)

i Then, using the method similar to Theorem 13, we can conclude that V σ (Tk ,kk ) is convergent with marginal value γ as long as the switched signals satisfy (4.32). Then, (4.29) will hold if i < 0 hold.

Theorem 15 For the given scalars 0 < ηi < 1, t1 + t2  1, t1 > 0, t2 > 0,if there exist symmetric positive definite matrices i ∈ R(n+2l)×(n+2l) ,matrices Y1i , Y2i ∈ Rm×(n+2l) ,and scalars εai , εbi ,such that the following LMI holds, respectively: ⎤ ⎡ iT iT iT iT −ηt1 i 0 i AiT i E 1iT Y1iT E b i C 0 1 + Y1 Bb ⎥ ⎢ iT iT ⎢ ∗ −ηt2 i i A2 + Y2iT BbiT 0 Y2iT E b 0 0 ⎥ ⎥ ⎢ i iT ⎢ ∗ 0 0 0 Hi ⎥ ∗ −i + εai D1i D1iT + εbi D b D b ⎥ ⎢ ⎥ ⎢ ∗ ∗ −εai I i 0 0 0 ⎥ 0 and assume that all boundary conditions are equal to 0, the following inequations will be derived: N1  N2 

V (x(t, k)) < i

t0 k1 −Vhi (x h (t, k))



N2 

N1  N2  

Vhi (x h (t + 1, k)) + Vvi (xv (t + 1, k))

t0 k1

− Vvi (xv (t + 1, k − 1))

Vhi (x h (N1 , k))

+

N1 



Vvi (xv (t, N2 )) ≥ 0

(4.38)

t0

k1

Therefore, N1 N2

((γ i )−1 V i iT i (x(t, k)) − γ i VIii (wi (t, k)))

t0 k1 N1 N2

≤



t0 k1 N1 N2

C C

 V i (x(t, k)) + (γ i )−1 V i iT i (x(t, k)) −γ i VIii (wi (t, k)) C C

(4.39)

Jk (t, k) < 0

t0 k1

Namely, Z (t, k) < γ i w(t, k). So, the robust H∞ performance index is less than γ i . This is the end of the proof.

4.2.4 Performance Optimization Considering this case of wi (t, k)  0, it is necessary to take system’s H∞ issue into account and design the controller by Theorem 14. On this occasion, t convergence index, k convergence index, and 2D robust H∞ property index are considered to be the decision variables to be optimized, which is a nonlinear optimization problem. If t convergence index and k convergence index are known, the following optimization algorithm will be applied: max

i ,Y1i ,Y2i ,εai ,εbi ,γ i

ηi

[subject to (4.29) or (4.35)]

(4.40)

110

4 Iterative Learning Control of Multi-phase Batch Processes

Using fixed-step or variable-step search algorithms, problem (4.40) can be solved. Given the upper bounds of the expected convergence index 0 < t1 ηi , t2 ηi < 1, the following optimization issue is solved: Minimi ze γ i

i ,Y1i ,Y2i ,εai ,εbi

(4.41)  Subject to (4.36)  i (t1 ηi , t2 ηi )  diag t1 ηi ih t2 ηi iv > 0, i  diag{ih , iv } > 0, εai , εbi > 0. Using the LMI optimization toolbox in MATLAB, the problem described by (4.41) can be solved. If the above optimization problem has a workable solution, the iterative updating law can be designed. Remark 14 The average dwell time under repetitive disturbances and non-repetitive  ∗ i disturbances can be derived through Theorems 14 and 15, i.e., τia ≥ τia  − lnln μηi , where μi can be solved by (4.30) and (4.36). Similar to the Eq. (4.20a) of Theorem 13, we can get ηi of Theorem 14.

4.3 Iterative Learning Control for Multi-phase Batch Processes with Time Delay and Disturbances In the former parts, the control problem of multi-phase batch process without time delay is studied, and the following research is done for multi-phase batch process with time delay.

4.3.1 Problem Statement At each batch of the process operation, a process iP-delay is considered, which describes a discrete model with uncertain parameter perturbation and time-varying delay in a certain range i

: P-delay ⎧ ⎨ x(t + 1, k)  Aσ (t,k) x(t, k) + Adσ (t,k) x(t − d(t), k) + B σ (t,k) u(t, k) + ωσ (t,k) (t, k) y(t, k)  Cσ (t,k) x(t, k) ⎩ x(0, k)  x0,k 0 ≤ t ≤ T, k  0, 1, 2 . . . (4.42)

4.3 Iterative Learning Control for Multi-phase Batch …

111

For multi-phase batch processes, since all the states of each batch are separated into p phases, and for each phase i(i  1, 2 . . . p), the system state x i (t + 1, k) is represented as:

i i i x i (t + 1, k)  A x i (t, k) + Ad x(t − d(t), k) + B u i (t, k) + ωi (t, k) (4.43) y i (t, k)  C i x i (t, k)

4.3.2 Robust Hybrid 2D Iterative Learning Control Design Based on Time Delay in a Range 4.3.2.1

The Equivalent 2D Representation

Define: xei (t − d(t), k)  x i (t − d(t), k) − x i (t − d(t), k − 1)

(4.44)

According to (4.42), (4.7), (4.8a), and (4.44), we get i

i

i

xei (t + 1, k)  A xei (t, k) + Ad xei (t − d(t), k) + B r i (t, k) + ω¯ i (t, k) i

(4.45a)

i

ei (t + 1, k)  ei (t + 1, k − 1) − C i A xei (t, k) − C i Ad xe (t − d(t), k) i −C i B r i (t, k) − C i ω¯ i (t, k)

(4.45b)

where ω¯ i (t, k)  δ(Ai )x i (t, k − 1) + δ(Aid )x i (t − d(t), k − 1) + δ(B i )u i (t, k − 1) + δ(ωi (t, k))

(4.45c)

Then, the FM model can be rewritten as: ⎡

⎡ i ⎡ i ⎡ i ⎤ ⎤ ⎤ ⎤ xei (t + 1, k) xe (t, k) xe (t + 1, k − 1) xe (t − t(d), k) i i i ⎣ x˜ i (t + 1, k) ⎦  A1 ⎣ x˜ i (t, k) ⎦ + A2 ⎣ x˜ i (t + 1, k − 1) ⎦ + Ad ⎣ x˜ i (t − t(d), k) ⎦ ei (t + 1, k) ei (t, k) ei (t + 1, k − 1) ei (t − t(d), k) + B˜ i r i (t, k) + H i ω¯ i (t, k) (4.46) where ⎡

i A1

⎤ ⎡ ⎤ 0 0 Ai Di  ⎣ 0 I i I i ⎦ + ⎣ 0 ⎦ F i (t, k) E i 0 0  A˜ i1 + D˜ i F i (t, k) E˜ i −C i Ai 0 0 −C i D i

112

4 Iterative Learning Control of Multi-phase Batch Processes

⎡

⎤ ⎤ ⎡ Aid 0 0 Di  i Ad  ⎣ 0 0 0 ⎦ + ⎣ 0 ⎦ F i (t, k) E di 0 0  A˜ id + D˜ i F i (t, k) E˜ di −C i D i −C i Aid 0 0 ⎡ ⎤ ⎡ ⎤ Bi Di i B  ⎣ 0 ⎦ + ⎣ 0 ⎦ F i (t, k)E bi  B˜ bi + D˜ i F i (t, k) E˜ bi −C i B i −C i D i ⎡ ⎡ i ⎤ ⎤ 00 0 I  i i i ⎣ ⎣ ⎦ A2  0 0 0 , H  0 ⎦, C  0 0 I i −C i 0 0 Ii Because the system is continuous before switching, the state at the moment of switching can be designed as (4.25). According to (4.46), the updating law can be designed as (4.26a). The system model for phase i, (i  1, 2 . . . p) is: ⎡ i ⎡ i ⎤ ⎤ ⎤ xei (t + 1, k)  xe (t, k)  xe (t + 1, k − 1)  i  i i i ⎣ x˜ i (t + 1, k) ⎦  A1 + B K 1i ⎣ x˜ i (t, k) ⎦ + A2 + B K 2i ⎣ x˜ i (t + 1, k − 1) ⎦ ei (t + 1, k) ei (t, k) ei (t + 1, k − 1) ⎡ i ⎤ xe (t − d(t), k) i + Ad ⎣ x˜ i (t − d(t), k) ⎦ + H i ω¯ i (t, k) (4.47) i e (t − d(t), k) ⎡

⎡

⎤ xe (t, k)  For ⎣ x(t, ˜ k) ⎦  x(t, k), considering (4.47), it can be rewritten as: e(t, k) ⎧ 2D−P-delay−C :  σ (t,k)  σ (t,k)  σ (t,k)  ⎪ ⎨ x(t + 1, k)  A1k x(t, k) + A2k x(t + 1, k − 1) + Adk x(t − d(t), k) + H σ (t,k) ω(t, ¯ k) ⎪ σ (t,k)  ∧ ⎩ x(t, k) Z (t, k)  e(t, k)  C (4.48) i Definition  boundary condition and ω¯ (t, k)  0, define χ   8 For any system with   ¯ sup x t,k : t + k  z, t¯, k ≥ 1 , if D ≥ z ≥ 0 and there are positive constants      a, b, γ such that for any x t,k ∈ Rn×n , the solution to system (4.48) satisfies x t,k  ≤ D #   b (D−z)    γ x ; then, the closed-loop 2D system (4.48) is with exponential stability   t,k a z

when the switching signal σ (·, ·) is met.

4.3 Iterative Learning Control for Multi-phase Batch …

4.3.2.2

113

Robust Hybrid ILC Based on Time Delay

Theorem 16 Assume ω¯ i (t, k) ≡ 0 holds. For given scalars 0 ≤ dm ≤ d M , ρ¯i 1, ξai , ξbi < 1 such that the following LMIs hold, respectively ⎡

i11 ⎢ ∗ ⎢ ⎣ ∗ ∗

i12

i22 ∗ ∗

i13 0

i33 ∗

⎤

i14 0 ⎥ ⎥ 0, the inequality (4.51) leads to 

V i (t + 1, k + 1)  V i (T0 + i, K 0 + 1)

t+kT0 +K 0 +i+1 T0 0 and some scalars 0 ≤ dm ≤ d M , let i αˆ i P1i + βˆi P2i < P i where αˆ i > 1, βˆi > 1, the 2D control law r i (t, k)  K 1i x (t, k) + i

K 2i x (t + 1, k − 1) is a robust H∞ guaranteed cost control law for system (5.55) if there exist symmetric positive matrices P i , P1i , P2i , Q i , W i and R i ∈ R(n+l)×(n+l) , scalar εai , εbi , u i > 0 and a positive scale ρ¯i 1, βˆi > 1, the robust guaranteed cost control problem of the 2D system (5.55) is i solvable if symmetric positive matrices, Q˜ i , Q¯ i , R˜ i , W˜ i , W , X i , i ∈ R(n+l)×(n+l) , matrices Y1i , Y2i ∈ Rm×(n+l) , and positive scalars ξai , ξbi , ρ¯i 1, βˆi > 1, the robust H∞ guaranteed cost control problem of the 2D system (5.55) is solvable if symmetric positive matrices i i , Q˜ i , Q¯ i , R˜ i , W˜ i , W , X i ∈ R(n+l)×(n+l) , matrices Y1i , Y2i ∈ Rm×(n+l) , and positive scalars ξai , ξbi , ρ¯i 0 z(t, k) S ⎡ ⎤ X Y1 Y2 ⎣ Y T P 1 0 ⎦ ≥ 0 with X ≤ rm2 1 Y2T 0 P 2

(6.8)

(6.9)

⎤ Z C1 ( A11 S + B1 Y1 ) C1 ( A2 S + B1 Y2 ) C1 E 1 0 ⎢ ( A S + B Y )T C T P1 0 0 S F1T ⎥ ⎥ ⎢ 11 1 1 1 ⎥ ⎢ 0 P2 0 0 ⎥≥0 ⎢ ( A2 S + B1 Y2 )T C1T ⎥ ⎢ ⎣ 0 0 λ−1 I 0 ⎦ E 1T C1T 0 0 λI 0 F1 S ⎡

with Z ≤ δym2 (6.10) where S  γ r −1 P −1 , P 1  γ r −1 P −1 P1 P −1 , P 2  γ r −1 P −1 P2 P −1 , and choose the gain of control law (6.5) as follows: H1  r γ −1 Y1 P,

H2  r γ −1 Y2 P

(6.11)

Proof Due to δw(t, k) ≡ 0, combining (6.4) and (6.5), we get ⎧ z(t + i + 1|t , k + j + 1|k ) ⎪ ⎪ ⎨  ( A1 + B1 H1 )z(t + i|t , k + j + 1|k ) + ( A2 + B1 H2 )z(t + i + 1|t , k + j|k )   ⎪ e(t + i|t , k + j|k ) ⎪ ⎩ δy(t + i|t , k + j + 1|k )   C1 z(t + i|t , k + j + 1|k ) e(t + i|t , k + j + 1|k ) (6.12) Define a Lyapunov function

6.2 2D-FM Model-Based Robust Iterative Learning Predictive …

193

V [z(t + i|t , k + j|k )]  VT [z(t + i|t , k + j|k )] + VK [z(t + i|t , k + j|k )]  z T (t + i|t , k + j|k )P1 z((t + i|t , k + j|k )) + z T (t + i|t , k + j|k )P2 z(t + i|t , k + j|k )

(6.13)

where P1 > 0 and P2 > 0. Assume that P1 + P2 < P, if the system is to be asymptotically stable, then the following constraint should be satisfied: V [z(t + i + 1|t , k + j + 1|k )]  VT [z(t + i + 1|t , k + j + 1|k )] − VT [z(t + i|t , k + j + 1|k )] + VK [z(t + i + 1|t , k + j + 1|k )] − VK [z(t + i + 1|t , k + j|k )]  z T (t + i + 1|t , k + j + 1|k )(P1 + P2 )z(t + i + 1|t , k + j + 1|k )

T



z(t + i|t , k + j + 1|k ) P1 0 z(t + i|t , k + j + 1|k ) − z(t + i + 1|t , k + j|k ) z(t + i + 1|t , k + j|k ) 0 P2 < z T (t + i + 1|t , k + j + 1|k )P z(t + i + 1|t , k + j + 1|k )





T P1 0 z(t + i|t , k + j + 1|k ) z(t + i|t , k + j + 1|k ) − z(t + i + 1|t , k + j|k ) z(t + i + 1|t , k + j|k ) 0 P2

z T (t + i|t , k + j + 1|k )Q 1 z(t + i|t , k + j + 1|k ) + z T (t + i + 1|t , k + j|k )Q 2 ≤− z(t + i + 1|t , k + j|k ) + r T (t + i|t , k + j + 1|k )Rr (t + i|t , k + j + 1|k )

(6.14) Making a summation of above inequality from i, j  0 to ∞, then ∞ ∞  

V [z(t + i + 1|t , k + j + 1|k )]

j0 i0

⎛

⎞ VT [z(t + 1|t , k + j + 1|k )] − VT [z(t|t , k + j + 1|k )] ⎜ +V [z(t + 1|t , k + j + 1|k )] − V [z(t + 1|t , k + j|k )] ⎟ K ⎜ K ⎟ ∞ ⎜ ⎟  ⎜ +VT [z(t + 2|t , k + j + 1|k )] − VT [z(t + 1|t , k + j + 1|k )] ⎟ ≤ ⎜ ⎟ ⎜ +VK [z(t + 2|t , k + j + 1|k )] − VK [z(t + 2|t , k + j|k )] ⎟ j0 ⎜ ⎟ ⎝ + . . . + VT [z(t + ∞ + 1|t , k + j + 1|k )] − VT [z(t + ∞|t , k + j + 1|k )] ⎠ +VK [z(t + ∞ + 1|t , k + j + 1|k )] − VK [z(t + ∞ + 1|t , k + j|k )] ≤

∞ 

{VT [z(t + ∞ + 1|t , k + j + 1|k )] − VT [z(t|t , k + j + 1|k )]}

j0

+

∞ 

{VK [z(t + i + 1|t , k + ∞ + 1|k )] − VK [z(t + i + 1|t, k|k )]}

i0

≤−

∞ ∞   j0 i0

J (t + i|t , k + j+1|k )

(6.15)

194

6 Iterative Learning Predictive Control for Batch Processes

For 2D system (6.4), assume that it has a finite set of initial conditions; i.e., there exist two positive integers i, j, such that z(t + i, k)  0, i ≥ r1 ; z(t, k + j)  0,

j ≥ r2

where r1 < ∞ and r2 < ∞ are positive integers. Here, z(t + i, k) and z(t, k + j) are called K-boundary and T -boundary at current time and current batch, respectively. Choose r  max{r1 , r2 }, then ∞ ∞  

V [z(t + i + 1|t , k + j + 1|k )]

j0 i0

≤

∞ 

{VT [z(t + ∞ + 1|t , k + j + 1|k )] − VT [z(t|t , k + j + 1|k )]}

j0

+

∞ 

{VK [z(t + i + 1|t , k + ∞ + 1|k )] − VK [z(t + i + 1|t , k|k )]}

i0 r1 

≤−

VT [z(t|t , k + j + 1|k )] −

j0

r2 

VK [z(t + i + 1|t , k|k )]

i0

≤ −r1 VT [z(t|t , k + 1|k )] − r2 VK [z(t + 1|t , k|k )] ≤ −r V [z(t|t , k|k )]  −r V [z(t, k)] ≤ −

∞ ∞  

J (t + i|t , k + j + 1|k )

j0 i0

i.e. J∞ (t, k + 1) 

∞ ∞  

J (t + i|t , k + j + 1|k ) ≤ r V [z(t, k)] ≤ γ

(6.16)

j0 i0

where γ is the upper bound of J∞ (t, k + 1). From V [z(t, k)] ≤ z(t, k)T Pz(t, k) ≤ r −1 γ  γ1 , the form (6.8) obviously holds. According to (6.12) and (6.13), (6.14) can be rewritten as:   T  T  ( A1 + B1 H1 )T ( A1 + B1 H1 )T z(t + i|t , k + j + 1|k ) P ( A2 + B1 H2 )T ( A2 + B1 H2 )T z(t + i + 1|t , k + j|k )      T   T T    z(t + i|t , k + j + 1|k ) Q1 0 H1 H1 P1 0 + + R − ≤0 H2T H2T 0 P2 0 Q2 z(t + i + 1|t , k + j|k ) (6.17) Obviously, (6.17) is true if the following is established

6.2 2D-FM Model-Based Robust Iterative Learning Predictive …



195



T







T (A1 + B1 H1 )T (A1 + B1 H1 )T Q1 0 H1T H1T P1 0 P + + R − ≤0 (A2 + B1 H2 )T (A2 + B1 H2 )T H2T H2T 0 P2 0 Q2

(6.18) Note that P  γ1 S −1 , thus ⎡

⎤ 1/2 γ1−1 P1 0 AT1 + H1T B1T Q 1 0 H1T R 1/2 1/2 ⎢ 0 γ1−1 P2 AT2 + H2T B1T 0 Q 2 H2T R 1/2 ⎥ ⎢ ⎥ ⎢ ⎥ S 0 0 0 ⎢ A1 + B1 H1 A2 + B1 H2 ⎥ ⎢ ⎥≥0 ⎢ Q 1/2 ⎥ 0 0 γ I 0 0 1 1 ⎢ ⎥ 1/2 ⎣ ⎦ 0 0 γ1 I 0 0 Q2 R 1/2 H2 0 0 0 γ1 I R 1/2 H1

(6.19)

Multiply (6.19) from the right by diag{S, S, I, I, I, I } and from the left by its transpose, and define Yi  Hi S, P i  γ1−1 S Pi S, i  1, 2, and then (6.19) is equivalent to ⎡

1/2

P1 0 S AT1 + Y1T B1T S Q 1 ⎢ 0 P2 S AT2 + Y2T B1T 0 ⎢ ⎢ S 0 ⎢ A1 S + B1 Y1 A2 S + B1 Y2 ⎢ ⎢ Q 1/2 S 0 0 γ 1I 1 ⎢ 1/2 ⎣ 0 0 0 Q2 S R 1/2 Y2 0 0 R 1/2 Y1

⎤ 0 Y1T R 1/2 1/2 S Q 2 Y2T R 1/2 ⎥ ⎥ ⎥ 0 0 ⎥ ⎥≥0 0 0 ⎥ ⎥ γ1 I 0 ⎦ 0 γ1 I

(6.20)

where I is the unity matrix with appropriate dimensions. Based on A1  A11 + E(t, k)F, Schur’s lemma and Lemma 1, (6.20) is equivalent to (6.6). Since Y1 , Y2 and S can be derived from the solution of (6.6), the control law can be solved as follows: H1  Y1 S −1  γ1−1 Y1 P,

H2  Y2 S −1  γ1−1 Y2 P

For the input and output constraints in (6.4), they will be represented as (6.9) and (6.10) if there exist a scalar λ > 0, and symmetric matrices X and Z. Theorem 32 Assume δw(t, k + 1)  0 holds. Consider the system (6.4) with structured uncertainty and given positive matrices Q 1 , Q 2 ∈ R (n+l)×(n+l) , R ∈ R m×m , and scalars rm > 0, δym > 0, the robust MPC problem is feasible if there exist symmetric positive definite matrices P1 , P2 , P ∈ R (n+l)×(n+l) , matrices Y1 , Y2 ∈ R m×(n+l) , symmetric matrices X ∈ R m×m and Z ∈ R l×l , and scalars γ1 > 0, μ > 0, ε > 0, λ > 0 such that the following LMIs hold

196

6 Iterative Learning Predictive Control for Batch Processes

⎡

P1 0 ⎢ 0 P2 ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ A1 S + B1 Y1 A2 S + B1 Y2 ⎢ ⎢ Q 1/2 S 0 ⎢ 1 1/2 ⎢ 0 Q ⎢ 2 S ⎢ 0 0 ⎢ ⎢ ⎢ R 1/2 Y1 R 1/2 Y2 ⎢ ⎣ 0 0 FS 0

1/2

0 S AT11 + Y1T B1T S Q 1 0 S AT2 + Y2T B1T 0 εI DT 0 D S 0 0 0 γ1 I 0 0 0 0 0 0 0 0 0 T 0 E 0 0 0 0

0 1/2 S Q2 0 0 0 γ1 I 0 0 0 0

T⎤ 0 Y1T R 1/2 0 S F ⎥ 0 Y2T R 1/2 0 0 ⎥ ⎥ 0 0 0 0 ⎥ ⎥ 0 0 E 0 ⎥ ⎥ 0 0 0 0 ⎥ ⎥≥0 0 0 0 0 ⎥ ⎥ ⎥ 0 0 0 ⎥ γ1 I ⎥ 0 γ1 I 0 0 ⎥ ⎥ 0 0 μ−1 I 0 ⎦ 0 0 0 μI

(6.21) where ε  γ −1 η and choose the gain of control law (6.5) as (6.11). Proof Theorem 32 can be obtained using lines similar to those in the proof of Theorem 31.

6.3 2D Fuzzy Constrained Predictive Control of Nonlinear Batch Processes In this chapter, a series of studies have been carried out on a linear constrained batch processes in the presence of disturbances and uncertainty. In contrast, the study of nonlinear batch processes with constraints is lagging behind. Based on this, the following research has been proposed.

6.3.1 Question Description and Modeling For batch processes with nonlinearity and uncertainty, an effective description will be a 2D T-S model as follows: ⎧ r r   ⎪ ⎪ ⎪ h f (z(t, k))A f x(t, k) + h f (z(t, k))B f u(t, k) + ω(t, k) ⎨ x(t + 1, k)  ⎪ ⎪ ⎪ ⎩

f 1

y(t, k)  C x(t, k) x(0, k)  x0,k ,0 < t < T p ,  |u(t, k)| ≤ u¯ and |y(t, k)| ≤ y

f 1

f  1, 2, . . . , r ; g  1, 2, . . . , p

(6.22) where u¯ and y express the input upper bound of the input constraint value and the actually upper bound output constraint value, respectively.

6.3 2D Fuzzy Constrained Predictive Control of Nonlinear Batch Processes

197

6.3.2 Design of 2D T-S Fuzzy Iterative Learning Predictive Controller 6.3.2.1

Equivalent 2D Description

Based on (6.22), we have δ(x(t + 1, k)) 

r 

h f (z(t, k))( A f δ(x(t, k)) + B f r (t, k)) + w(t, k)

(6.23)

f 1

e(t + 1, k)  yr (t + 1) − y(t + 1, k)  e(t + 1, k − 1) − C

r 

h f (z(t, k))( A f x(t, k)+B f r (t, k)) − Cw(t, k)

f 1

(6.24) where w(t, k)  w(t, ˜ k) + δ(ω(t, k)), δ(ω(t, k))  ω(t, k) − ω(t, k − 1), w(t, ˜ k) 

r  f 1

δ(h f (z(t, k)))A f x(t, k − 1) +

r 

δ(h f (z(t, k)))B f u(t, k − 1).

f 1

Then, based on (6.23) and (6.24), an equivalent 2D error model is obtained: ⎧



r  ⎪ x h (t + 1, k) x h (t, k) ⎪ ⎪ h f (z(t, k))A f  ⎪ ⎪ ⎪ xv (t, k + 1) xv (t, k) ⎪ f 1 ⎪ ⎪ ⎪ ⎪ r ⎨   h f (z(t, k))B f r (t, k) + Dw(t, k) + (6.25) 2D−BP : ⎪ ⎪ f 1 ⎪ ⎪ ⎪

⎪ ⎪ ⎪ x h (t, k) ⎪ ⎪ Z (t, k)  e(t + 1, k − 1)  C ⎪ ⎩ x (t, k) v

       δ(x(t, k)) Af 0 x h (t, k)  , C  0 I , Af  ,Bf  where e(t + 1, k − 1) xv (t, k) −C A f I     Bf I ,D  and I represents the unity matrix with proper dimensions; −C B f −C x h (t, k) ∈ R n 1 , xv (t, k) ∈ R n 2 represent horizontal and vertical state components of proper dimension vector, and Z (t, k) represents the system’s output. This model is an error model of system (6.22) in essence, which may equivalently represent the

198

6 Iterative Learning Predictive Control for Batch Processes

dynamic behavior of system (6.22). Therefore, the design of the updating law r (t, k) for system (6.22) is equivalent to the design of the control law for this system. Then, the closed-loop fuzzy model is represented as follows: 

:

2D−BP−C

⎧ r r   ⎪ ⎪ ⎨ x (t, k)  h f (x(t, k))A f x(t, k) + h f (x(t, k))B f r (t, k) + Dw(t, k) ⎪ ⎪ ⎩

f 1

f 1

z(t, k)  e(t + 1, k − 1)  Cx(t, k) (6.26)

where x (t, k) 



x h (t + 1, k) xv (t, k + 1)



 

   δk (x(t + 1, k)) δk (x(t, k)) , x(t, k)  . e(t + 1, k) e(t + 1, k − 1)

Over infinite horizon [t, ∞), [k, ∞), a “worst-case” performance index of a uncertain system at time instant t in phase k is defined as follows: max

min

r (t+i|t ,k+ j|k ),i, j0,1,...,∞ w(t,k)2 ≤z(t,k)2

J∞ (t, k) 

∞  ∞ 

J (t + i|t , k + j|k )

i0 j0

T    ∞  ∞   x(t + i|t , k + j|k ) x(t + i|t , k + j|k ) Q 0  r (t + i|t , k + j|k ) 0 R r (t + i|t , k + j|k )

(6.27)

i0 j0

subject to ⎧ r  ⎪ ⎪ ⎪ x (t + i|t , k + j|k )  h f (z(t, k))A f x(t + i|t , k + j|k ) ⎪ ⎪ ⎪ ⎪ f 1 ⎪ ⎪ ⎪ r ⎪  ⎪ ⎪ ⎨ + h f (z(t, k))B f r (t + i|t , k + j|k ) f 1 ⎪ ⎪ ⎪ ⎪ ⎪ Dw(t + i|t , k + j|k ) + ⎪ ⎪ ⎪ ⎪ ⎪ y˜ (t + i|t , k + j|k )  C y x(t + i|t , k + j|k ) ⎪ ⎪ ⎪ ⎩ z(t + i|t , k + j|k )  C x(t + i|t , k + j|k)



r (t + i|t , k + j|k )2 ≤ u¯ y(t + i|t , k + j|k )2 ≤ y

(6.28)

(6.29)

where u¯ and y are the bounds for the input and output variables. Q(Q > 0) and R(R > 0) are weighting matrices with proper dimensions.

6.3 2D Fuzzy Constrained Predictive Control of Nonlinear Batch Processes

199

The following 2D-ILC iterative updating law is introduced to make this system reach quadratic stability, which is designed as follows: r (t + i|t , k + j|k ) 

r  f 1

 h f (z(t, k))K i

δ(x(t + i|t , k + j|k )) e(t + i + 1|t , k + j − 1|k )

 (6.30)

The control objective of ILMPC problem is to design (6.30) to minimize the cost function J∞ (t, k) subject to model uncertainty and input/output constraints as described in (6.28), (6.29). Remark 24 In view of the nonlinear batch processes, most results did not take the system output tracking and the constraints into consideration. In order to achieve this target, this section will first describe such problems and then propose the corresponding controller design to make the system operate steadily, which is obviously different from traditional fuzzy control methods due to the proposed method can deal with problem in both time and batch directions. Remark 25 (6.27) is a “min-max” optimization problem, in which “max” is used to search the largest or “worst-case” value of J∞ under uncertainty. The “min” is used to search for the current and future control variable in order to minimize this worst case. The “min-max” problem is not easy to handle under the infinite horizon MPC. It is to design r (t + i|t , k + j|k ) to minimize a “worst-case” infinite horizon performance index (6.27). By using linear matrix inequality (LMI) theory, it can be transformed into a convex optimization with LMI constrains. Then, only the current control law r (t|t , k|k ) is realized at discrete-time t and the optimization problem is repeated at discrete-time t + i in phase k. The system stability is proved by 2D Lyapunov, which is defined as follows: V (x(t + i|t , k + j|k ))  θ · x T (t + i|t , k + j|k)M −1 x(t + i|t , k + j|k )  Vh (x h (t + i|t , k + j|k )) + Vv (xv (t + i|t , k + j|k )), i, j  0, . . . , ∞ (6.31) where M > 0 ∀k, t ≥ 0. For the model (6.27)–(6.29), the following conditions must be satisfied: (a) Inequality constraint of 2D Lyapunov function: x T (t + i|t , k + j|k )T Qx(t + i|t , k + j|k ) + r T (t + i|t , k + j|k )Rr (t + i|t , k + j|k ) ≤ V (x(t + i|t , k + j|k )) − V (x (t + i|t , k + j|k )) (6.32) (b) For 2D system (6.29), assume that it has a finite set of initial conditions; i.e., there exist two positive integers i, j, such that

200

6 Iterative Learning Predictive Control for Batch Processes

x(t + i, k)  0, i ≥ m 1 ; x(t, k + j)  0,

j ≥ m2

where n 1 < ∞ and n 2 < ∞ are positive integers. Here, x(t + i, k) and x(t, k + j) are called K-boundary and T -boundary at current time and current batch, respectively. Formula (6.32) is made a summation of above inequality from i, j  0 to ∞ to draw the following inequality: max J∞ (t, k)  m 1 V [x h (t, k)] − m 2 V [xv (t, k)] ≤ mV [x(t, k)] ≤ θ

6.3.2.2

(6.33)

Stability Analysis and Controller Design of a 2D System

Theorem 33 Given semi-positive definite symmetric matrices R, Q, the ILMPC problem is feasible if there exist a positive definite symmetric matrix M  h

Q 0 v , matri0 Q ces Y f , Yg ( f  1, 2, . . . , r ; g  1, 2, . . . , p), scalars ε f , εg , γ , θ > 0 and 0 < σ < 1, 0 < μ < 1, such that the following matrix inequalities hold: diag{ M h , M v } , a semi-positive definite symmetric matrix Q 

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

−M(σ, μ) + (s − 1)Q ∗ ∗ ∗ ∗ ∗ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

1

1

−γ In 0 0 0 ∗ −θ1 In ∗ ∗ −θ1 In ∗ ∗ ∗ ∗ ∗ ∗

−M(σ, μ) − Q 0 ∗ −γ In ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 

T

T

Y Tf R 2 M T Q 2 M A f + Y Tf B f M T C

T YT f +Yg 2

1 2

1 2

T

D 0 0 −M ∗

0 0 0 0 −γ In T

R M T Q ψ1 M T C 0 0 0 0 0 0 0 −θ1 In 0 ∗ −θ1 In 0 ∗ ∗ −M 0 ∗ ∗ ∗ −γ In  T −1 x (t|t , k|k ) ≤0 ∗ −M   −u¯ 2 I Y f ≤0 ∗ −M 

−y 2 M MC T ∗ −I

T⎤

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ 0, there exists the controller (6.46) such that (6.43) is feasible if there exist symmetric positive matrices P1s , P2s , P s ∈ R (n+l)×(n+l) , symmetric matrices X s ∈ R m×m and Z ∈ R l×l , and scalars μs > 0, 0 < ρ¯s < 1, αˆ s , βˆ s > 1, such that the following LMIs hold ⎡

⎤  s 1/ 2 sT sT − P˜1s 0 AsT Q1 0 H1sT (R s )1/ 2 1 + H1 B1  s 1/ 2 sT s 1 2 ⎥ ⎢ sT sT 0 Q2 H2 (R ) / ⎥ ⎢ ∗ − P˜1s AsT 2 + H2 B1 ⎢ ⎥ −1 s s ⎢ ∗ ⎥ ∗ −γ1 (P ) 0 0 0 ⎢ ⎥ 0, (6.43) is feasible if there exist symmetric positive matrices s s P 1 , P 2 , S s ∈ R (n+l)×(n+l) , matrices Y1s , Y2s ∈ R m×(n+l) , and scalars γ1s > 0, μs > 0 such that the following LMIs hold ⎤  s 1/2 s sT sT s −P 1 0 S s AsT 0 Y1sT (R s )1/2 1 + Y1 B1 S Q 1  1/2 sT s 1/2 ⎥ s ⎢ sT sT 0 S s Q s2 Y2 (R ) ⎥ ⎢ ∗ −P 1 S s AsT 2 + Y2 B1 ⎥ ⎢ ⎥ ⎢ ∗ ∗ −S s 0 0 0 ⎥ 0 and α i  0 refer to partial failure and complete failure. If the system has stuck or completely failed fault, the controller will not function. Therefore, here we only consider αi > 0. Remark 28 From equality (7.7) and the definition of α0 , it is not difficult to see that each of the diagonal elements in α0 should satisfy the following case: −βi0 ≤ α0i ≤ βi0 , (0 ≤ i ≤ m). Thus, in view of (7.5c), inequality (1 − βi0 )βi ≤ (1 + α0i )βi ≤ α¯ −α α¯ +α α¯ −α α¯ +α (1 + βi0 )βi holds. Thus, (1 − α¯ii +α i ) i 2 i ≤ (1 + α0i )βi ≤ (1 + α¯ii +α i ) i 2 i . This is i i obviously equal to equality (7.3). This is the reason that why (7.6) is derived from (7.4a–7.4d) and (7.5a–7.5c).

7.2.2 Robust 2D FILRC Design 7.2.2.1

Equivalent 2D Representation

Introducing the input error and the state error, an augmented model is obtained:

218

7 Iterative Learning Fault-Tolerant Control …

 2D-P-delay-F ⎧

:

⎪ ⎨ x(t + 1, k)  ( A1 + Aa (t, k))x(t, k) + A2 x(t + 1, k − 1) + ( Ad + Ad (t, k))x(t − d(t), k) + Bαr (t, k) + Dω(t, k) ⎪ ∧ ⎩ z(t, k)  e(t, k)  Gx(t, k)

(7.8)





δk (x(t + 1, k)) A 0 00 , A1  where x(t + 1, k)  , A2  , Ad  e(t + 1, k) CA 0 0I



  Ad 0 B I ¯ ¯ , B  k) F, , D  , G  0 I , A(t, k)  E(t, C Ad 0 CB C

    E ¯ k) F¯d , E¯  Ad (t, k)  E(t, , F¯  F 0 , F¯d  Fd 0 . Therefore, CE it is called the equivalent 2D model of the ILC system. It is distinct from the uncertain 2D-FM systems in that the information propagation in the time direction only occurs over a finite duration. However, this specificity does not prevent the difference from the general 2D-FM uncertainty model to discuss the robust FTC problem of 2D system ( 2D-P-delay-F ). Thus, based on the 2D theory, the FILRC law can be designed. Design a control law as follows: 

:

2D-ec-delay-F

r (t, k)  K 1 x(t, k) + K 2 x(t + 1, k − 1)

(7.9)

Then, the closed-loop 2D-FM fault system is given by 

:

2D-P-delay-F-C

⎧ ⎪ x(t + 1, k)  ( A1 + Bα K 1 + Aa (t, k))x(t, k) + ( A2 + Bα K 2 )x(t + 1, k − 1) ⎪ ⎪ ⎪ ⎪ ⎨ + ( Ad + Ad (t, k))x(t − d(t), k) + Dω(t, k) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

 A˜ 1F (t)x(t, k) + A˜ 2F x(t + 1, k − 1) + A˜ d (t)x(t − d(t), k) + Dω(t, k) ∧

z(t, k)  e(t, k)  Gx(t, k) (7.10)

 For 2D system 2D-P-delay-F-C , the state evolves along two axes called the T -axis and the K-axis. Obviously, the boundary conditions of the 2D system should be of two dimensions, which are denoted by

7.2 Robust Delay-Dependent Iterative Learning … h x(t, k)  xt,0 , ∀0 ≤ t < r1 , x(t, k)  0, t ≥ r1 , d x(t, k)  x0,k , ∀0 ≤ k < r2 , x(t, k)  0, k ≥ r2 , d h  x0,0 x0,0

219

k  −h M , −h M + 1, · · · , 0, k  −h M , −h M + 1, · · · , 0, t  −d M , −d M + 1, · · · , 0, t  −d M , −d M + 1, · · · , 0,

(7.11)

h d where r1 < ∞ and r2 < ∞ are positive integers, and xt,0 and x0,k are given vectors d h called the K-boundary and T -boundary, respectively. If x0,0  x0,0  0, then they are called the zero boundary conditions.

7.2.2.2

Reliable Controller Design and System Structure

In this section, the main objective is to establish a robust delay-related sufficient condition for a class of FILRC laws using the LMI technique (7.10) and to give a method to obtain the reliable controller. The results are as follows. Theorem 36 Assume ω(t, k) ≡ 0 for given scalars 0 ≤ dm ≤ d M , 0 < λ < 1; a delay-range-dependent sufficient condition for an existing 2D reliable controller that guarantees the closed-loop 2D system (7.10) to be robustly stabilizable is that ¯ W , W˜ , X ∈ R (n+l)×(n+l) , matrices Y1 there exist symmetric positive matrices L, Q, m×(n+l) , Y2 ∈ R , and positive scalars εi (i  1, 2) such that the following LMI is established ⎡

11 ⎢ ∗ ⎢ ⎣ ∗ ∗

12 22 ∗ ∗

13 23 33 ∗

⎤ 14 24 ⎥ ⎥ 0. To prove  < 0, first, pre- and postmultiply (7.12) by diag[P, P, P, R, I , I, I, I, I, I ] ; define L  P −1 , X  R −1 ,  ¯ ˜ L W L  W , L Q L  Q, X W X  W , Y  Y1 Y2  K 1 L K 2 L ; let R¯ 1 replace L 1 R1 L 1 , respectively. Then, using Schur complements and Lemma 2, the inequality ⎡ ⎤ 11 12 13 14 ⎢ ∗ 22 23 24 ⎥ ⎥ (7.12) is transformed into inequality (7.16), that is, ⎢ ⎣ ∗ ∗ 33 0 ⎦ < 0 is ⎡

∗ ∗ ∗ 44 equivalent to  < 0. Hence, V (t + 1, k) < 0 holds for any ϕ(t, k)  0, and the following inequality is effective: V (t + 1, k) ≤ Vh (t, k) + Vv (t + 1, k − 1)

(7.17)

Using the normal system approach, the faulty system is progressively stable.  Thus, based on definition 1, the closed-loop 2D state delay system 2D-P-delay-F-C is 2D fault-tolerant. Theorem 37 For some given scalars 0 ≤ dm ≤ d M , γ > 0, if there exist symmetric ¯ W , W˜ , X ∈ R (n+l)×(n+l) , matrices Y1 , Y2 ∈ R m×(n+l) , and positive matrices L, Q, positive scalars εi (i  1, 2) such that the following LMI holds ⎡ ⎤ 11 12 13 14 15 ⎢ ∗ ⎥ ⎢ 22 23 24 0 ⎥ ⎢ ⎥ (7.18) ⎢ ∗ ∗ 33 0 0 ⎥ < 0 ⎢ ⎥ ⎣ ∗ ∗ ∗ 44 0 ⎦ ∗ ∗ ∗ ∗ −I



  11 0 12 T T T with , 13  , 11    D D D 12 21 0 −γ 2 I



21  T 13 14 15 , 14  , 15  with 15  G L 0 0 0 0 , and others are 0 0 0  designed in Theorem 36; then, the closed-loop 2D state delay system 2D-P-delay-F-C is 2D controllable via a 2D reliable control law and has

an H∞ performance less than   x(t, k) γ ; the r (t, k)  Y1 L −1 Y2 L −1 can be called a γ -suboptimal x(t + 1, k − 1) 2D reliable state feedback H∞ controller. where

222

7 Iterative Learning Fault-Tolerant Control …

Proof To establish the H∞ performance of the 2D system (7.10) with zero boundary conditions for any nonzero ω(t, k) ∈ l2 {[0, ∞], [0, ∞]}, introduce J

∞  ∞ 

[z T (t, k)z(t, k) − γ 2 ωT (t, k)ω(t, k)]

(7.19)

t0 k0

In view of the stability of the system and the zero initial condition, and for any nonzero ω(t, k) ∈ l2 {[0, ∞], [0, ∞]} J≤

∞  ∞ 

[z T (t, k)z(t, k) − γ 2 ωT (t, k)ω(t, k) + V (t + 1, k)]

(7.20)

t0 k0

However, z T (t, k)z(t, k) − γ 2 ωT (t, k)ω(t, k) + V (t + 1, k) ⎛    T    T ⎞

T T T T T 1 1 1 1 ⎠ ϕ(t, k) ϕ(t, k) ⎝ 2  1 + + d P R T T T T 1 M ω(t, k) ω(t, k) D D D D (7.21) ⎡

11 + G T G ⎢ 0 ⎢ ⎢ where  1  ⎢ 0 ⎢ ⎣ R 0 Thus, it follows that

0 −P 0 0 0

0 0 −Q 0 0

⎤ R 0 0 0 ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ 12 0 ⎦ 0 −γ 2 I

⎡ ⎛ ⎤    T    T ⎞

T

∞  ∞ T T T T      ϕ(t, k) ϕ(t, k) 2 1 P 1 1 R 1 ⎣ ⎝ 1 + ⎠ ⎦ J≤ + dM T T T T ω(t, k) ω(t, k) D D D D t0 k0

(7.22) Using the inequality (7.18) with the proof of Theorem 36, LMI (7.18) is clearly equivalent to  1 +

  T    T T1 T1 T1 T1 2 + dM 0, a delay-dependent sufficient condition for an  existing FILRC controller that guarantees the closed-loop 2D system 2D-P-delay-F-C   (7.35) to be robustly stabilizable for any time-varying delay d ∈ 0, d¯ is that there ¯ Q, ˜ X ∈ R (n+l)×(n+l) , matrices Y1 , Y2 ∈ should be symmetric positive matrices L, Q, m×(n+l) , and positive scalars εi (i  1, 2)such that the following LMI holds R ⎡ 11 ⎢ ∗ ⎢ ⎣ ∗ ∗

 12  22 ∗ ∗

 13 23 33 ∗

 14 ⎤ 24 ⎥ ⎥ 0, γ > 0, if there exist symmetric positive ¯ Q, ˜ X ∈ R (n+l)×(n+l) , matrices Y1 , Y2 ∈ R m×(n+l) , and positive scalars matrices L, Q, εi (i  1, 2) such that the following LMI holds ⎡− → − → − → − → − → ⎤ 11 12 13 14 15 ⎢  22 23 24 0 ⎥ ⎢ ∗ ⎥ ⎢ ⎥ (7.27) ⎢ ∗ 0 ⎥ 0, Q > 0, W > 0, W > 0, X > 0, ˜ ˜ ≥ 0, L X





¯ ¯ R I L I R I (7.28) ≥ 0, ≥ 0, ≥ 0, εi >0 (i  1, 2). I R˜ I L˜ I X˜

7.3 Delay-Range-Dependent Method for Iterative Learning …

225

7.3 Delay-Range-Dependent Method for Iterative Learning Fault-Tolerant Guaranteed Cost Control for Batch Processes The first part is to establish a 2D-FM model for batch processes with non-repetitive disturbance, time delay, and actuator failure. However, in the actual production process, it is necessary not only to ensure the robust stability of the closed-loop system, but also to ensure a certain robust performance. Therefore, the cost control method is widely used, which not only makes the system stable, but also guarantees a good performance level in a system with uncertain parameters. Therefore, the following research has been conducted for the cost control of batch processes with guaranteed performance.

7.3.1 Equivalent 2D Representation Use a concept similar to 7.2, a batch process with state delay and actuator failures can be described by 

:

P-delay-f

⎧ ⎨ x(t + 1, k)  A(t, k)x(t, k) + Ad (t, k)x(t − d(t), k) + Bu F (t, k) y(t, k)  C x(t, k) ⎩ x(0, k)  x(0); ≤ t ≤ T ; k  1, 2, . . .

(7.29)

For (7.29), the control objective is to design a reliable guaranteed cost control law such that the output of the process tracks a given trajectory yd (t) as closely as possible even with actuator failures and the system preserves the H∞ performance index and the least guaranteed cost. For the purpose of achieving steady-state tracking error along the time direction with fast convergence, it is assumed here that the general model of the feedback/feedforward control to be extended is described by the linear dynamic model as follows: 

: xm (t + 1, k)  Am xm (t, k) + Bm e(t, k)

(7.30)

m

where xm (t, k) is an extended state dynamically determined by e(t, k); {Am , Bm } are parameters that are specified based on the structure of the feedback and/or feedforward controls to be extended. In this paper, we simply specify that Am  Bm  I . By introducing the state errors, the output errors, and the extended information, an augmented model is obtained:

226

7 Iterative Learning Fault-Tolerant Control …



:

2D-P-delay-f



x(t + 1, k)  A1 (t, k)x(t, k) + A2 x(t + 1, k − 1) + Ad (t, k)x(t − d(t), k) ∧ + Bαr (t, k) + Dω(t, k)z(t, k)  e(t, k)  Gx(t, k) (7.31)

where ¯ ¯ A1 (t, k)  A1 + Aa (t, k)  A1 + E(t, k) F, ¯ Ad (t, k)  Ad + Ad (t, k)  Ad + E(t, k − 1) F¯d , ⎡

⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ⎤ A 00 000 Ad 0 0 B A1  ⎣ 0 I I ⎦, A2  ⎣ 0 0 0 ⎦, Ad  ⎣ 0 0 0 ⎦, B  ⎣ 0 ⎦, C Ad 0 0 CA 0 0 00I CB ⎡ ⎤ ⎡ ⎤ I E     E¯  ⎣ 0 ⎦, F¯  F 0 0 , D  ⎣ 0 ⎦, G  0 0 I . C CE T  In view of control design, the state variable x(t, k)  xeT (t, k) xmT (t, k) eT (t, k) in (7.31) is to be stabilized to final steady-state point of the original state space. z(t, k) indicates measured tracking error in the current cycle to be minimized against load disturbance. The above model is a typical 2D model with parameter uncertainty and state delay. Since it equivalently represents the dynamic characteristics of the system tracking error (7.29), it is called the equivalent tracking error of system (7.29). Therefore, it is obvious that the design of the updating law r (t, k) is equivalent to  the design of a fault-tolerant control law for the equivalent 2D tracking error model 2D-P-delay-F . Remark 29 Note that the internal perturbations of system (7.28) have been converted into the external perturbations if ω(t, k)  0. It is well known that ILC methods can successfully solve repeated operational information in a batch process. However, they cannot maintain the robust stability of a time-invariant process model system. Therefore, it is necessary to consider new design methods. Thus, the feedback is proposed in conjunction with the iterative learning control design methods. To analyze the sensitivity of the controlled output to the disturbance, we thereby take into account the robust H∞ control problem since the robust H∞ performance indicates the upper bound of the sensitivity of the controlled output to the disturbance. The following cost function is associated with the 2D system (7.31) J

N1  N2   t0 k0

 x T (t, k)U1 x(t, k) + x T (t, k − 1)U2 x(t, k − 1) + r T (t, k)U3r (t, k)

7.3 Delay-Range-Dependent Method for Iterative Learning …



N1  N2 

ϕ T (t, k)

t0 k0

U1 + K 1T U3 K 1 K 1T U3 K 2 ϕ(t, k) K 2T U3 K 1 U2 + K 2T U3 K 2

227

(7.32)

  where ϕ T (t, k)  x T (t, k) x T (t + 1, k − 1) , U1 > 0, U2 > 0, and U3 > 0. Remark 30 For the partial failure case, the authors proposed an iterative learning reliable control (ILRC) scheme for batch processes without time delay case in [1]. In this paper, the author described the traditional design approach that only improves the tracking performance in the time direction. Since the design objective for batch processes requires to improve the control performance not only along the time direction but also along the batch direction. Then, the closed-loop fault-tolerant system is given by 

:

2D-P-delay-f-C



x(t, k)  A˜ 1F x(t, k) + A˜ 2F x(t + 1, k − 1) + A˜ d x(t − d(t), k) + Dω(t, k) ∧ z(t, k)  e(t, k)  Gx(t, k) (7.33)

where A˜ 1F  A1 (t, k) + Bα K 1 , A˜ 2F  A2 + Bα K 2 , A˜ d  Ad (t, k). The following definitions and lemmas will be introduced to establish the design procedure for the updating law r (t, k) to ensure that the system is both robust and stable and preserves an adequate control performance.

7.3.2 ILRG Design In this section, we will design a reliable guaranteed cost updating law r (t, k) such that the resulting system (7.33) is 2D fault-tolerant guaranteed cost control and the cost function of the system is below the specified upper bound. Theorem 38 For some given scalars 0 ≤ dm ≤ d M , t, and t1 , the robust faulttolerant H ∞ guaranteed cost control problem of the closed-loop 2D system (7.33) is ¯ X ∈ R (n+l)×(n+l) , ¯ Q, ˜ W ,W˜ , R, solvable if there exist symmetric positive matrices L, Q, matrices Y1 , Y2 ∈ R m×(n+l) , and positive scalars εi (i  1, 2) and γ > 0 such that the following LMI holds ⎤ ⎡ 11 12 13 14 15 ⎢ ∗ 0 0 0 ⎥ ⎥ ⎢ 22 ⎥ ⎢ (7.34) ⎢ ∗ ∗ −γ I 0 0 ⎥ < 0 ⎥ ⎢ ⎣ ∗ ∗ ∗ 44 0 ⎦ ∗ ∗ ∗ ∗ 55

228

7 Iterative Learning Fault-Tolerant Control …

where ⎤ ⎡ T T T T T ⎤ Y1 L A1 + Y1T β B L A1 + Y1T β B − L L 0 φ11 0 0 T T T T ⎥ ⎢ T ⎢ 0 φ12 0 0 0 ⎥ L A2 + Y2T β B ⎥ ⎢ Y2 L A2 + Y2T β B ⎥ ⎢ ⎥ ⎢ T T ⎥ ˜ ⎢  , 0 0 ⎥ 12 ⎢ 0 ⎥, L A L A ⎢ 0 0 −Q d d ⎥ ⎢ ⎣ L 0 0 −W˜ − X 0 ⎦ ⎦ ⎣ 0 0 0 T T 0 0 0 0 −γ I 0 D D ⎤ ⎤ ⎡ ⎡ ⎤ ⎡ T LG T L L 0 Y1 β L F¯ T ⎢ 0 ⎥ ⎢0 0 L⎥ ⎢ Y Tβ 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎢ ⎥ ¯T ⎥  ⎢ 0 ⎥, 14  ⎢ 0 0 0 ⎥, 15  ⎢ ⎢ 0 L Fd ⎥, ⎣ 0 ⎦ ⎣0 0 0⎦ ⎣ 0 0 ⎦ 0 0 0 0 0 0 ⎤ ⎡ −1 0 0 −U3 ⎦,  ⎣ 0 −t L +   0  −d2 X +  ⎡

11

13

22

  ¯ 44  −diag W , U1−1 , U2−1 , 55  −diag[ε1 I, ε2 I ], with φ11  −t1 L +d1 Q˜ − R, T −2 φ12  −(t − t1 )L , d1  d M − dm + 1,   ε1 Bβ02 B + ε2 E¯ E¯ T , d2  d M .

In this case, for any delay d(t) satisfying 0 ≤ dm ≤ d(t) ≤ d M , the reliable H ∞ guaranteed cost control law can be designed as r (t, k)  Y1 L −1 x(t, k) + Y2 L −1 x(t + 1, k − 1)

(7.35a)

 and the corresponding cost function of the resulting closed-loop 2D system 2D-P-delay-f-C (7.33) satisfies   2 J ≤ r2 t1 β + d M γ1 + 21 (d M + dm )(d M − dm )γ1 +d M γ2 + 21 d M (d M + 1)γ3 + r1 (t − t1 )β  J ∗ (7.35b) Proof First, we prove asymptotical stability of the closed-loop 2D system. This part can be obtained by using lines similar to those in the proof of Theorem 37. Next, we will prove that the robust H ∞ performance level in equality z 2 D−2e ≤ γ ω 2 D−2e is satisfied for ω(t, k)  0. This part can be obtained using lines similar to those in the proof of Theorem 19. Finally, we will prove that equality (7.35b) holds. It is easy to obtain that N1  N2  t0 k1

V ≤

N1  N2  t0 k1

U1 + K 1T U3 K 1 K 1T U3 K 2 ϕ(t, k) (7.36) −ϕ (t, k) K 2T U3 K 1 U2 + K 2T U3 K 2 T

For N1 , N2 → ∞, it follows from equality (7.36), definition 1, boundary conditions, and the definitions of P1 and P that

7.3 Delay-Range-Dependent Method for Iterative Learning … N1  N2  t0 k1

229

N2  U1 + K 1T U3 K 1 K 1T U3 K 2 ϕ(t, k) ≤ ϕ (t, k) V (0, k) K 2T U3 K 1 U2 + K 2T U3 K 2

T

k1

+

N1 

V (t + 1, 0) (7.37)

t0

Thus, it follows from equality (7.37) and boundary conditions that J≤

N2 

[x T0,1 t1 L −1 x 0,1 +

k0

+ dM

−1 

T ¯ −1 Q x l,1 + x l,1

l−d(0) −1 −1   s−d M ls

T ηl,1 X −1 ηl,1 ] +

−1 

T x l,1 W

l−d M N1 

−1

x l,1 +

−d m

−1 

T ¯ −1 Q x xl,1 x l,1

s−d M ls

x T1,0 (t − t1 )L −1 x 1,0

t0

  2 ≤ r2 t1 β + d M γ1 + 21 (d M + dm )(d M − dm )γ1 +d M γ2 + 21 d M (d M + 1)γ3 + r1 (t − t1 )β  J∗

(7.38)

−1 where β  λmax (L −1 ), γ1  λmax ( Q¯ −1 ), γ2  λmax (W ), γ3  λmax (X −1 ). This completes the proof of Theorem 38.

Remark 31 The use of Lemma 4 in Theorem 38 avoids complex calculation and more constrained conditions, which leads to new and less conservative delay-rangedependent stable criteria. From equalities (7.34) and (7.35b), It can be seen that the sizes of the parameters d M , dm , t, and t1 directly affect not only whether the above theorem is solvable or not, but also when the corresponding guaranteed cost function and the H ∞ performance index achieve the least value. Therefore, these parameters must be adjusted to be of appropriate sizes. The specific steps can be seen in the following algorithm. Remark 32 Note that the sizes of β and β0 are associated with the upper and lower ¯ of α, which also affect whether the above theorem is solvable or bounds (α and α) not. The choice of values of α and α must be considered. The reason is that the choice of the sizes of α and α¯ is considered not only based on the above element, but also based on the practical factor. For example, in injection molding, the hydraulic control valve opening, which is considered to be the actuator, cannot be too small (that is, α is sufficiently large) because it not only affects the nozzle packing pressure but also indirectly affects product quality. Remark 33 As can be seen from equality (7.34), it establishes conditions that depend on the upper and lower bounds of the time delay. This differs from the results in the related literature in that the delay is usually fixed. In fact, for inequality (7.34), choose d M  dm ; that is, in the definition of the Lyapunov function, let Q  0, and the interval delay can be degraded to a constant delay. Therefore, the results in this paper are general and include the case of constant delays as a special case.

230

7 Iterative Learning Fault-Tolerant Control …

7.3.3 Design Algorithms In order to obtain robust fault-tolerant H ∞ guaranteed cost controller r (t, k)  Y1 L −1 x 0,1 + Y2 L −1 x 1,0 and achieve as far as possible the least guaranteed cost value J ∗ , we have to solve the following optimization problem   2 minr2 t1 β + d M γ1 + 21 (d M + dm )(d M − dm )γ1 +d M γ2 + 21 d M (d M + 1)γ3 + r1 (t − t1 )β (7.39)





−β I T −γ1 I T −γ2 I T < 0, < 0, 0, (7.34),

Note that in the matrix inequality (7.34), the conditions are no longer LMI con˜ L X −1 L  R, ¯ and L W −1 L  W˜ . We ditions because of the terms L Q¯ −1 L  Q, introduce an identity matrix I and transform them into LMIs, which are shown in (7.40). On the other hand, from the conditions in (7.40), it can be seen that the values of t and t1 will affect the bound of the cost. So, we adopt the following steps: Given larger t and t1 , solve the inequality (7.40). If there is a feasible solution, then given smaller t and t1 , and go on; otherwise, stop until we obtain the least guaranteed cost value J ∗ and the corresponding least H ∞ performance index γ .

7.4 Design of Fault-Tolerant Guaranteed Performance Controller for Batch Process Compound Iterative Learning The research for 7.2 and 7.3 is based on state feedback. This method is based on a fact that states are measurable. When states are unmeasurable, the control of the system will not be realized. Therefore, this section proposes a control method that combines output feedback and ILC, which overcomes the shortcomings of state feedback, better achieves the expected control objectives, and yields good control effects.

7.4.1 Equivalent Two-Dimensional Description In order for the system output to quickly track a given output, more learning information can be utilized. Therefore, the following extended information is introduced.

7.4 Design of Fault-Tolerant Guaranteed Performance Controller …



ˆ k) + Ae2 e(t, k) x e (t + 1, k)  Ae1 x e (t, k) + Ade x e (t − d, ye (t, k)  C e1 x e (t, k) + De1 e(t, k)

231

(7.41)

where xe (t, k) is the structural parameter of the appropriate information dimension determined by e(t, k) and the appropriate dimension given by {Ae , Be }. From (7.1) to (7.7), the following dimensionality model is obtained: ⎧ x1 (t + 1, k)  ( A1 + Δ¯ a (t, k))x1 (t, k) + A2 x1 (t + 1, k − 1) ⎪ ⎪ ⎪ ⎪ ⎨ ˆ k) + Bαr (t, k) + H¯ ω(t, k) + ( A + Δ¯ (t, k))x (t − d, d

⎪ ⎪ ⎪ ⎪ ⎩

d

1

y(t, k)  x1 (t, k)  C x 1 (t, k) ¯ 1 (t, k) z(t, k)  e(t, k)  Gx

(7.42)

where ⎡

⎡ ⎡ ⎤ ⎤ ⎤ δ(x(t + 1, k)) A 0 0 000 x1 ⎣ xe (t + 1, k) ⎦, A1  ⎣ 0 Ae1 Ae2 ⎦, A2  ⎣ 0 0 0 ⎦, CA 0 0 e(t + 1, k) 00I ⎡ ⎡ ⎤ ⎤ Ad 0 0 B Ad  ⎣ 0 Ade 0 ⎦, B  ⎣ 0 ⎦, C Ad 0 0 CB   ¯ ¯ k) F¯a , Δ¯ d (t, k)  EΔ(t, k) F¯d , F¯a  F 0 0 , Δ¯ a (t, k)  EΔ(t, ⎡ ⎡ ⎤ ⎤ ⎡ ⎤ E −C 0 I I     F¯d  Fd 0 0 , G¯  0 0 I , E¯  ⎣ 0 ⎦, H¯  ⎣ 0 ⎦, C ⎣ 0 Ce1 De1 ⎦. 0 0 I CE C Equation (7.42) is an equivalent 2D model of the ILC system. The equivalent expression (7.1) is equivalently expressed as a specific 2D faulty model, and at the same time it can describe the dynamic characteristics of the tracking and convergence of the ILC system. Based on the 2D theory, the following composite iterative learning reliable performance controller (ILRGCC) is designed: ⎧ ˆ ⎪ ⎨ xc (t + 1, k)  Ac1 xc (t, k) + Ac2 xc (t + 1, k − 1) + Ac3 xc (t − d, k) + Bc1 y(t, k) + Bc2 y(t + 1, k − 1) ⎪ ⎩ r (t, k)  Cc1 xc (t, k) + Cc2 xc (t + 1, k − 1) + Dc1 y(t, k) + Dc2 y(t + 1, k − 1) (7.43) n where xc (t, k) ∈ R is the internal state of the controller and controller parameter Aci , Bcj , Ccj , Dcj i1,2,3; j1,2 is pending. Then, the closed-loop 2D system is as follows:

232

7 Iterative Learning Fault-Tolerant Control …

⎧ a (t, k))x(t, x(t ˜ + 1, k)  ( A˜ 1 + Δ ˜ k) + A˜ 2 x(t ˜ + 1, k − 1) ⎪ ⎪ ⎪ ⎪ ⎨ ˆ k) + H˜ ω(t, k)  (t, k))x(t + ( A˜ + Δ ˜ − d, d

⎪ ⎪ ⎪ ⎪ ⎩

d

y(t, k)  C˜ x(t, ˜ k)

(7.44)

z(t, k)  e(t, k)  G˜ x(t, ˜ k)

where

Ai x1 (t + 1, k) ˜ x(t ˜ + 1, k)  , Ai  xc (t + 1, k)

Ad 0 A˜ d  , 0 Ac3



 Δ¯ a (t) 0 E¯ ˜ Δa (t)   Δ(t, k) F¯a 0 0 0



¯  Δd (t) 0 E¯ Δ˜ d (t)   Δ(t, k) F¯d 0 0 0     C˜  C 0 , G˜  G¯ 0 .

+ Bα Dci C BαCci , (i  1, 2), Bci C Aci

 ˜ ˜ k) F, 0  EΔ(t,  T  ˜ k) F˜d , H˜  H¯ T 0 , 0  EΔ(t,

where M¯  diag[M0]. Combined with (7.44), the following performance function is given J

N1  N2  

x˜ T (t, k)U1 x(t, ˜ k) + x˜ T (t + 1, k − 1)U2 x(t ˜ + 1, k − 1)

t0 k1 T

 + r (t, k)U3r (t, k)

N1  N2  K 1T U3 K 2 U + K TU K ϕ(t, k) ϕ T (t, k) 1 T 1 3 1  K 2 U3 K 1 U2 + K 2T U3 K 2

(7.45)

t0 k1

where   U1 > 0, U2 > 0, U3 > 0, ϕ T (t, k)  x˜ T (t, k) x˜ T (t + 1, k − 1) ,   K i  Dci C Cci .

7.4.2 Iterative Learning Reliable Guaranteed Performance Control Law Design Theorem 39 Given 0 ≤ dm ≤ d M and positive numbers, 0 < αi < 1, t > 1, t1 + t4 < 1, t2 , and 0 < αi < 1, t > 1, t1 + t4 < 1, t2 , closed-loop systems (7.44),

7.4 Design of Fault-Tolerant Guaranteed Performance Controller …

233

the guaranteed cost control is solvable if there are matrices X > 0, Y > 0, Dci , Z i , Z¯ i , Z˜ 1 , and Zˆ i (i  1, 2) and positive numbers ε, λ, and γ that make the following inequalities true ⎡

⎤ 11 12 13 ⎣ ∗ 22 0 ⎦ < 0 ∗ ∗ 33

(7.46a)

where ⎡

11

13

⎡ ⎤ −2 0 −d M t J P 0 J¯A1 t J A2 t J Ad 0 t J H¯ ⎢ ⎢ ⎥ ∗ −J P J A1 J A2 J Ad 0 J H¯ ⎥ ⎢ ⎢ 0 ⎢ ⎢J ⎥ ∗ ∗ Π1 0 0 tJP 0 ⎥ ⎢ ⎢ P ⎢ ⎢ ⎥ ⎢ 0 0 ⎥, 12  ⎢ 0 ∗ ∗ ∗ −α1 Q˜ 2 0 ⎢ ⎢ ⎥ ⎢ ⎢ 0 ∗ ∗ ∗ ∗ − Q˜ 0 0 ⎥ ⎢ ⎢ ⎥ ˜ ⎣ ⎣ 0 ⎦ ∗ ∗ ∗ ∗ ∗ Wp 0 ∗ ∗ ∗ ∗ ∗ ∗ −γ I 0 ⎡ ⎤ 0 t JE¯ 0 t J¯B 0 ⎢ 0 JE¯ 0 J¯B 0 ⎥ ⎢ ⎥ ⎤ ⎡ ⎢ J T 0 J T 0 J¯T ⎥ 0 −U1−1 0 ¯ ⎢ F¯1 ⎥ K 1 G ⎢ ⎥  ⎢ 0 0 0 0 J¯KT 2 ⎥, 22  ⎣ ∗ −U2−1 0 ⎦, ⎢ T ⎥ ⎢J¯ 0 0 0 0 ⎥ ∗ ∗ −U3−1 ⎢ F1 ⎥ ⎣ 0 0 0 0 0 ⎦ 0

0

0

⎤ 0 0 ⎥ ⎥ JKT1 ⎥ ⎥ ⎥ JKT2 ⎥, ⎥ 0 ⎥ ⎥ 0 ⎦ 0

0

⎤ −λI 0 0 0 0 ⎢ ∗ −λ−1 I 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ∗ ∗ −γ I 0 0 ⎥ ⎥ ⎢ ⎣ ∗ ∗ ∗ −ε I 0 ⎦ ∗ ∗ ∗ ∗ −ε−1 I ⎡

33

0

0 0 0 JP 0 0 0

˜ 1 + α2 Q ˜ 2 < JP α1 Q

(7.46b)

where ˜ J¯A  t(J A − J P ), W˜ p  −W˜ − t J P , 1  −α1 Q˜ 1 + W˜ − t J P + (d M − dm + 1) Q, 1

1

X I Zˆ i S Ai + Z¯ i C T T , J¯K i  β JK i , J¯B  J B β0 , J p  , J Ai  I Y Ai +Bβ D ci C Ai Y + Bβ Z i    



F¯ T F¯dT SB X Ad Z˜ 1 (i  1, 2), J B  , JFT¯  , JFT¯   , J T T , J E¯  A d 1 d B Ad Ad Y Y F¯ Y F¯ d

      X E¯ ¯ , JK i  Dci C Z i . , J HT¯  H¯ T X H¯ T , JG¯  G¯ GY E¯ Moreover, the system output matrix Aci , Ac3 , Bci , Cci ,and Dci can be solved

234

7 Iterative Learning Fault-Tolerant Control …

⎧ −1 T T −1 ⎪ ⎪ Aci  Aci  P¯12 ( Zˆ i − X Ai Y − Z¯ i CY − X BβCci P12 )(P12 ) ⎪ ⎪ −1 T −1 ⎪ ) ⎨ Ac3  P¯12 ( Z˜ 1 − X Ad Y )(P12 −1 ¯ ¯ Bci  P12 ( Z i − X Bβ Dci ) ⎪ ⎪ T −1 ⎪ Cci  (Z i − Dci CY )(P12 ) ⎪ ⎪ ⎩D  D ci ci

(7.46c)

The corresponding closed-loop system performance upper bound is satisfied as J ≤ J∗ 

N2  

x1T (0, k)t4 X x 1 (0, k) +

k1

−1

Σ

r −d(0)

x1T (r, k)t2 X x 1 (r, k)

−1

+ Σ x1T (r, k)t3 X x 1 (r, k) r −d M

−1

+ dM Σ



−1

Σ [x1 (r + 1, k) − x1 (r, k)] t X [x1 (r + 1, k) − x1 (r, k)] T

s−d M r s

+

−dm

Σ

−1

Σ x1T (r, k)t2 X x 1 (r, k) +

s−d M r s

N1 

x1T (t + 1, 0)t1 X x 1 (t + 1, 0)

t0

(7.46d) Proof Theorem 39 can be obtained using lines similar to those in the proof of Theorems 36 and 38.

7.4.3 Design Algorithm   2 minr2 t4 + d M t2 + 21 (d M + dm )(d M − dm )t2 + d M t3 + 2d M (d M + 1)t η + r1 t1 η

−ηI Θ T X Subject to t, t1 , t2 , t3 , t4 > 0 , t1 + t4 < 1, (7.46a) < 0, Θ X −X



−λ − 1 1 −ε − 1 1 < 0, 0, Q  0 Qv such that the following matrix inequalities (LMIs) establishes ⎤ ⎡ T T Ω Ai + YiT β T B i YiT β T Ω1 ⎥ ⎢ T (8.4) ⎣ Ai Ω + B i βYi −Ω + εi Bi α02 B i 0 ⎦ < 0 0 −εi I βYi ⎡ T T T T T⎤ −Ω(σ,μ) − Q Ω2 Y j β Yi β ⎢ Ω2 Ω22 0 0 ⎥ ⎥ 0, a matrix H i ∈ R m×(n 1 +n 2 ) , scalars εai , εbi > 0 and γ i > 0 make the following inequalities holds: ⎤ iT i i iT −S i (β1i , β2i ) S i AiT Si E H i T β i Si G i T 0 2 + H β B2 ⎥ ⎢ 0 0 C2i ⎥ ∗ −S i + εai D2i D2iT + εbi B2i β0i2 B2i T 0 ⎢ ⎥ ⎢ i i ⎢ ∗ ∗ −εa I 0 0 0 ⎥ ⎥ 0 and assume that all boundary conditions are equal to 0, the following inequations will be derived: N2 N1  

V i (x(t, k))
0, εai , εbi > 0. If the above optimization problem has can be solved, the iterative updating law can be designed.

9.3 A Hybrid 2D Fault-Tolerant Controller Design for Multi-phase Batch Processes with Time Delay 9.3.1 Equivalent 2D Expression Based on Eq. (4.43) and gain fault, a multi-phase batch process with actuator faults can be described as follows: 

i

i

i

x i (t + 1, k)  A x i (t, k) + Ad x i (t − d(t), k) + B u iF (t, k) + ωi (t, k) y i (t, k)  C i x i (t, k)

(9.36)

278

9 Iterative Learning Fault-Tolerant Control of Multi-phase Batch …

By introducing the output errors and iterative learning control laws, their equivalent 2D-FM model may be adapted to the following form by equality (9.35): 

:

2D−P−delay−F

⎧ i i i i i i ⎪ ⎨ x (t + 1, k)  ( A1 + Aa (t, k))x (t, k) + A2 x (t + 1, k − 1) i i i i +( Ad + Ad (t, k))x i (t − d(t), k) + B α i r i (t, k) + D ω¯ i (t, k) ⎪ ⎩ i z (t, k)  ei (t, k)  G i x i (t, k) (9.37)

     i i i δ(x i (t + 1, k)) Ai 0 Aid 0 , A1  , Ad  ,B  x (t + 1, k)  i i i i i e (t + 1, k) −C A 0 −C Ad 0      i    i i i i i Bi 0 0 I , A2  ,D  , G i  0 I i , Aa (t, k)  E i (t, k)F , −C i B i −C i 0 Ii    i  i  i  i i i i i Ei Ad (t, k)  E i (t, k)F d , E  i i , F  F 0 , F d  Fd 0 . −C E Since the state of the system before switching is continuous, the switching state transition is similar to Sect. 4.2.2 of Chap. 4 and is omitted here. According to (9.37), the design of the updating law is as follows: 

where i

r i (t, k)  K 1i x i (t, k) + K 2i x i (t + 1, k − 1)

(9.38)

The closed-loop model in the phase i(i  1, 2 . . . q) can be described as follows: 

:

2D−P−delay−F−C

 i  ⎧ i i i i i ⎪ x (t + 1, k)  A + B α K + A (t, k) x i (t, k) ⎪ 1 a 1 ⎪  i  ⎪ ⎪ i ⎪ ⎪ + A2 + B α i K 2i x i (t + 1, k − 1) ⎪ ⎪ ⎨ i i i +( Ad + Ad (t, k))x i (t − d(t), k) + D ω¯ i (t, k) ⎪ %i2 x i (t + 1, k − 1) + A %id (t)x i (t − d(t), k) %i1 (t)x i (t, k) + A ⎪ A ⎪ ⎪ ⎪ i i ⎪ ⎪ +D ω¯ (t, k) ⎪ ⎪ ⎩ i ∧ i z (t, k)  e (t, k)  G i x i (t, k)

(9.39)

The switching system of the above model is: ⎧ %σ (t,k) %σ (t,k) %σ (t,k) ⎪ ⎪ x(t + 1, k)  A1 x(t, k) + A2 x(t + 1, k − 1) + Ad x(t − d(t), k) ⎨ ⎪ ⎪ ⎩

+D

σ (t,k)

ω(t, ¯ k)

∧

z(t, k)  e(t, k)  G σ (t,k) x(t, k) (9.40)

The conclusions are as follows.

9.3 A Hybrid 2D Fault-Tolerant Controller Design for Multi-phase …

279

9.3.2 Design of Delay-Range-Dependent Robust Hybrid 2D Iterative Learning Fault-Tolerant Control Law Theorem 49 Assume ω¯ i (t, k) ≡ 0 holds. For some given scalars 0 ≤ dm ≤ d M , 0 < αˆ i < 1, 0 < βˆi < 1, 0 < λi < 1, if there exist symmetric positive matrices L i , i i % i and X i ∈ R (n+l)×(n+l) , matrices Y1i , Y2i ∈ R m×(n+l) and positive scalars Q,W ,W μi > 1, ε1i , ε2i > 0 that lead to the following ⎡

i11 ⎢ ∗ ⎢ ⎣ ∗ ∗

i12 i22 ∗ ∗

i13 i23 i33 ∗

⎤ i14 i24 ⎥ ⎥ 0 such that the following matrix inequality holds:

282

9 Iterative Learning Fault-Tolerant Control of Multi-phase Batch …

⎡

i

 ⎢ 11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

i

12 i22 ∗ ∗ ∗

i

13 i23 i33 ∗ ∗

i

14 i24 0 i44 ∗

⎤ i 15 ⎥ 0 ⎥ ⎥ 0 ⎥ 0, a delay-dependent sufficient condition of the existing FILRC controller to ensure that the closed-loop ' ¯ 2D system 2D−P−delay−F−C (9.51) is robust to any time-varying delay d ∈ [0, d], i

is that there should be symmetric positive matrices L i , Q and X i ∈ R (n+l)×(n+l) , matrices Y1i , Y2i ∈ R m×(n+l) and positive scalars μi > 1, ε1i , ε2i > 0 such that the following matrix inequality holds

9.3 A Hybrid 2D Fault-Tolerant Controller Design for Multi-phase …

⎡ ˜i ˜i 11 12 ⎢ ∗  ˜ i22 ⎢ ⎣ ∗ ∗ ∗ ∗

˜ i13  i23 i33 ∗

283

˜ i14 ⎤  i24 ⎥ ⎥ 1, ε1i , ε2i > 0 such that the following Y1 , Y2 ∈ R matrix inequality holds: ⎡ i  ⎢ 11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

i

12 ˜ i22  ∗ ∗ ∗

i

13 i23 i33 ∗ ∗

i

14 i24 0 i44 ∗

⎤ i 15 ⎥ 0 ⎥ ⎥ 1, 0 < t1i < 1, 0 < t i < 1 and 0 < αˆ i < 1, if matrices i

i i

i

i

Q , Q , R , W , W , X i and i ∈ R(n+2l)×(n+2l) , matrices Y1i , Y2i ∈ Rm×(n+2l) and positive number ξai , ξbi > 0 exist and enable the following inequality to hold: ⎤ ⎡ iT iT iT i iT iT iT iT iT iT 1 2 3 4 6 7 8 9 5 i ⎥ ⎢ ⎢ ∗ −U 0 0 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ ∗ ∗ Φ1i Φ2i 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ ∗ ∗ 0 0 0 0 0 ⎥ ∗ Φ3i 0 ⎥ ⎢ i ⎢ ∗ ∗ ∗ ∗ −W 0 0 0 0 0 ⎥ ⎥ ⎢ (9.59) ⎥