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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley - IEEE
- Sprache: Englisch
STATE FEEDBACK CONTROL AND KALMAN FILTERING WITH MATLAB/SIMULINK TUTORIALS Discover the control engineering skills for state space control system design, simulation, and implementation State space control system design is one of the core courses covered in engineering programs around the world. Applications of control engineering include things like autonomous vehicles, renewable energy, unmanned aerial vehicles, electrical machine control, and robotics, and as a result the field may be considered cutting-edge. The majority of textbooks on the subject, however, lack the key link between the theory and the applications of design methodology. State Feedback Control and Kalman Filtering with MATLAB/Simulink Tutorials provides a unique perspective by linking state space control systems to engineering applications. The book comprehensively delivers introductory topics in state space control systems through to advanced topics like sensor fusion and repetitive control systems. More, it explores beyond traditional approaches in state space control by having a heavy focus on important issues associated with control systems like disturbance rejection, reference tracking, control signal constraint, sensor fusion and more. The text sequentially presents continuous-time and discrete-time state space control systems, Kalman filter and its applications in sensor fusion. State Feedback Control and Kalman Filtering with MATLAB/Simulink Tutorials readers will also find: * MATLAB and Simulink tutorials in a step-by-step manner that enable the reader to master the control engineering skills for state space control system design and Kalman filter, simulation, and implementation * An accompanying website that includes MATLAB code * High-end illustrations and tables throughout the text to illustrate important points * Written by experts in the field of process control and state space control systems State Feedback Control and Kalman Filtering with MATLAB/Simulink Tutorials is an ideal resource for students from advanced undergraduate students to postgraduates, as well as industrial researchers and engineers in electrical, mechanical, chemical, and aerospace engineering.
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Table of Contents
Cover
Title Page
Copyright
Dedication
Author Biography
Preface
Note
Acknowledgments
List of Symbols and Acronyms
About the Companion Website
Part I: Continuous‐time State Feedback Control
1 State Feedback Controller and Observer Design
1.1 Introduction
1.2 Motivation for Going Beyond PID Control
1.3 Basics in State Feedback Control
1.4 Pole‐assignment Controller
1.5 Linear Quadratic Regulator (LQR) Design
1.6 Observer Design
1.7 State Estimate Feedback Control System
1.8 Summary
1.9 Further Reading
2 Practical Multivariable Controllers in Continuous‐time
2.1 Introduction
2.2 Practical Controller I: Integral Action via Controller Design
2.3 Practical Controller II: Integral Action via Observer Design
2.4 Drive Train Control of a Wind Turbine
2.5 Summary
2.6 Further Reading
Part II: Discrete‐time State Feedback Control
3 Introduction to Discrete‐time Systems
3.1 Introduction
3.2 Discretization of Continuous‐time Models
3.3 Input and Output Discrete‐time Models
3.4
‐Transforms
3.5 Summary
3.6 Further Reading
Problems
4 Discrete‐time State Feedback Control
4.1 Introduction
4.2 Discrete‐time State Feedback Control
4.3 Discrete‐time Observer Design
4.4 Discrete‐time Linear Quadratic Regulator (DLQR)
4.5 Discrete‐time LQR with Prescribed Degree of Stability
4.6 Summary
4.7 Further Reading
5 Disturbance Rejection and Reference Tracking via Observer Design
5.1 Introduction
5.2 Disturbance Models
5.3 Compensation of Input and Output Disturbances in Estimation
5.4 Disturbance‐Observer‐based State Feedback Control
5.5 Analysis of Disturbance‐Observer‐based Control System
5.6 Anti‐windup Implementation of the Control Law
5.7 Summary
5.8 Further Reading
Note
6 Disturbance Rejection and Reference Tracking via Control Design
6.1 Introduction
6.2 Embedding Disturbance Model into Controller Design
6.3 Controller and Observer Design
6.4 Practical Issues
6.5 Repetitive Control
6.6 Summary
6.7 Further Reading
Part III: Kalman Filtering
7 The Kalman Filter
7.1 Introduction
7.2 The Kalman Filter Algorithm
7.3 The Kalman Filter in Multi‐rate Sampling Environment
7.4 The Extended Kalman Filter (EKF)
7.5 The Kalman Filter with Fading Memory
7.6 Relationship between Kalman Filter and Observer
7.7 Summary
7.8 Further Reading
8 Addressing Computational Issues in KF
8.1 Introduction
8.2 The Sequential Kalman Filter
8.3 The Kalman Filter using
Factorization
8.4 Summary
8.5 Further Reading
Bibliography
Index
End User License Agreement
List of Tables
Chapter 5
Table 5.1 Controller action corresponds to choice of disturbance model
List of Illustrations
Chapter 1
Figure 1.1 Closed‐loop control system.
Figure 1.2 The variations of closed‐loop poles for
.
Figure 1.3 Closed‐loop output responses. Key: line (1)
; line (2)
; line (...
Figure 1.4 Closed‐loop control response. Key: line (1)
; line (2)
; line (...
Figure 1.5 Three springs and double mass system (Example 1.1).
Figure 1.6 Block diagram of state feedback control system.
Figure 1.7 State variable responses (Example 1.2).
Figure 1.8 Closed‐loop state response to state initial values (Example 1.6)....
Figure 1.9 Closed‐loop state response to state initial conditions (Example 1...
Figure 1.10 Closed‐loop state response to state initial conditions (Example ...
Figure 1.11 Closed‐loop state response to state initial values (Example 1.9)...
Figure 1.12 Closed‐loop state response to state initial values (Example 1.10...
Figure 1.13 Closed‐loop state response to state initial values. Key: line (1...
Figure 1.14 Comparison between the true and estimated state variables using ...
Figure 1.15 Estimation of sinusoidal signal from noisy measurement (Example ...
Chapter 2
Figure 2.1 Comparison of closed‐loop control system responses in presence of...
Figure 2.2 Comparison of closed‐loop response (Example 2.2). Key: line (1)...
Figure 2.3 Schematic diagram of a drum boiler (Åström and Bell [2000]).
Figure 2.4 Input and output disturbances. Top figure: input disturbance on t...
Figure 2.5 Output signal in presence of disturbance. Top figure: drum level ...
Figure 2.6 Control signal for disturbance rejection. Top figure: feedwater f...
Figure 2.7 Output signal (solid line) and reference signal (dashed‐line). To...
Figure 2.8 Control signal. Top figure: feedwater flow rate (
); middle figur...
Figure 2.9 Constrained state estimate feedback control for unit step respons...
Figure 2.10 Disturbance rejection in sugar mill control.
Figure 2.11 Illustration of two mass model for turbine (Perdana [2008]).
Figure 2.12 Reference signals for the angular velocity and position.
Figure 2.13 Closed‐loop responses of turbine and generator. Key: line (1) ou...
Figure 2.14 Closed‐loop responses of turbine and generator with load demand ...
Figure 2.15 Closed‐loop responses of turbine and generator with load demand ...
Chapter 3
Figure 3.1 Illustration of the bridges between continuous‐time and discrete‐...
Figure 3.2 Illustration of the unit circle on the complex plane (solid line)...
Figure 3.3 Discrete‐time sinusoidal signals with different sampling interval...
Figure 3.4 Eigenvalues of discrete‐time system (Example 3.6). Key: *
; o
;...
Figure 3.5 Discrete‐time higher order model (Example 3.9)
Chapter 4
Figure 4.1 Closed‐loop response (Example 4.2).
Figure 4.2 Measurement data (Example 4.3).
Figure 4.3 Comparison between
and
(Example 4.3). Key: line (1)
; line (...
Figure 4.4 Illustration of eigenvalues in discrete‐time (Example 4.4). Key: ...
Figure 4.5 Illustration of closed‐loop eigenvalues with
(Example 4.4).
Figure 4.6 Illustration of closed‐loop eigenvalues with
and
(Example 4.4...
Figure 4.7 Comparison of closed‐loop state response with and without prescri...
Figure 4.8 Comparison of closed‐loop control response with and without presc...
Figure 4.9 Illustration of closed‐loop eigenvalues of the error system (Exam...
Figure 4.10 Comparison of error response with and without prescribed degree ...
Chapter 5
Figure 5.1 Random walk signal.
Figure 5.2 Sinusoidal disturbance.
Figure 5.3 Output signals in presence of sinusoidal disturbance (Example 5.1...
Figure 5.4 Comparison of error signals with and without disturbance (Example...
Figure 5.5 State error response in presence of disturbance (Example 5.3).
Figure 5.6 Comparison of state errors with and without prescribed degree of ...
Figure 5.7 Closed‐loop control system response using disturbance observer (E...
Figure 5.8 A quadruple tank system.
Figure 5.9 Closed‐loop output response to reference and disturbance (Example...
Figure 5.10 Closed‐loop control response to reference and disturbance (Examp...
Figure 5.11 Closed‐loop response to reference and disturbance (Example 5.8)....
Figure 5.12 Frequency response of the controller and sensitivity (Example 5....
Figure 5.13 Closed‐loop response to reference change with constraints (Examp...
Figure 5.14 Disturbance signal and normalized Fourier coefficients (Example ...
Figure 5.15 Open‐loop output responses to input disturbances (Example 5.10)....
Figure 5.16 Closed‐loop response to disturbances with constraints (Example 5...
Figure 5.17 Hot air balloon system.
Chapter 6
Figure 6.1 Closed‐loop control system response using state estimate feedback...
Figure 6.2 Closed‐loop output response to reference and disturbance (Example...
Figure 6.3 Closed‐loop control response to reference and disturbance (Exampl...
Figure 6.4 Closed‐loop output response to reference and disturbance. Key: li...
Figure 6.5 Closed‐loop response to reference signal (Example 6.4). Key: line...
Figure 6.6 Closed‐loop response to disturbance signal (Example 6.4). Key: li...
Figure 6.7 Closed‐loop response to reference signal (Example 6.5). Key: line...
Figure 6.8 Closed‐loop response to disturbance signal (Example 6.5). Key: li...
Figure 6.9 Realization of continuous‐time repetitive control system.
Figure 6.10 Realization of discrete‐time repetitive control system.
Figure 6.11 Picture of the robot arm showing pick and place locations. This ...
Figure 6.12 The trajectory for a robotic arm for a complete period (Example ...
Figure 6.13 The trajectories for
and
axes (Example 6.6).
Figure 6.14 Reconstructed reference signal of
(Example 6.6). Key: line (1)...
Figure 6.15 Reconstructed reference signal of
(Example 6.6). Key: line (1)...
Figure 6.16 Reference tracking using repetitive controller (Case A).
Figure 6.17 Reference tracking using repetitive controller (Case B).
Figure 6.18 Reference tracking using repetitive controller (Case C).
Figure 6.19 Open‐loop response of mechanical system.
Figure 6.20 Block diagram of a food extruder.
Figure 6.21 Desired trajectory in
x‐y
plane.
Chapter 7
Figure 7.1 Piece‐wise constant signal (Example 7.1). Key: line (1) parameter...
Figure 7.2 Measured output data (Example 7.2).
Figure 7.3 Performance evaluation of Kalman filter (Example 7.2).
Figure 7.4 Evaluation of Kalman filter performance when
matrix is wrong (E...
Figure 7.5 Performance evaluation of Kalman filter with compensation for sen...
Figure 7.6 Estimate of the sensor bias (Example 7.4). Key: line (1) the esti...
Figure 7.7 Performance evaluation of Kalman filter with compensation for loa...
Figure 7.8 Comparison between true and estimated state without GPS measureme...
Figure 7.9 Trajectory of the moving object (Example 7.6). Key: line (1) the ...
Figure 7.10 Comparison between true and estimated state with GPS measurement...
Figure 7.11 Trajectory of the moving object (Example 7.6). Key: line (1) the...
Figure 7.12 Diagonal element of
(Example 7.6). Top figure:
; bottom figur...
Figure 7.13 Comparison between true and estimated state with two GPS measure...
Figure 7.14 Trajectory of the moving object (Example 7.7). Key: line (1) the...
Figure 7.15 Estimation results for EKF (Example 7.8).
and
.
Figure 7.16 Estimation error for EKF (Example 7.8).
and
.
Figure 7.17 Schematic of a double tank.
Figure 7.18 Measured input and output signals (Example 7.9).
Figure 7.19 Comparison between the estimated and true water tank levels (Exa...
Figure 7.20 Application of Kalman filter with fading memory (Example 7.10). ...
Figure 7.21 Output signals (Example 7.11).
Figure 7.22 Estimation results from Kalman filter (Example 7.11).
and
.
Figure 7.23 Estimation results from Kalman filter (Example 7.11).
and
.
Figure 7.24 Comparative studies between KF and observer (Example 7.11). Key:...
Chapter 8
Figure 8.1 Comparison between
and
(Example 8.3). Key: line (1)
; line (...
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
Author Biography
Preface
Acknowledgments
List of Symbols and Acronyms
About the Companion Website
Begin Reading
Bibliography
Index
End User License Agreement
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State Feedback Control and Kalman Filtering with MATLAB/Simulink Tutorials
Liuping Wang and Robin Ping Guan
This edition first published 2023© 2023 John Wiley & Sons Ltd
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Liuping Wang and Robin Ping Guan to be identified as the authors of this work has been asserted in accordance with law.
Registered OfficeJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
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Limit of Liability/Disclaimer of WarrantyMATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This work's use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
Library of Congress Cataloging‐in‐Publication Data is Applied for:
Hardback ISBN: 9781119694632
Cover Design: WileyCover Image: © fotoslaz/Shutterstock.com
ToJianshe
Author Biography
Liuping Wang, PhD, is a Professor of Control Engineering at RMIT University, Australia. She obtained her PhD from the Department of Control Engineering at the University of Sheffield, UK. Professor Wang gained substantial process control experience by working in the Chemical Engineering Department at the University of Toronto, Canada, and the Center for Integrated Dynamics at the University of Newcastle, Australia. She is the author of five books in systems and control.
Robin Ping Guan, PhD, obtained his Masters degree in Electrical Engineering from the University of Melbourne in 2014 and his PhD degree from RMIT University, Australia in 2019. He is a research fellow in RMIT University and is the co‐author of the book on state feedback control and Kalman filter.
Preface
Between PID Control and Model Predictive Control
Control systems are usually designed using one of the two types of mathematical models: transfer function model and state space model. The PID controller is an example of transfer function‐based design and is very popular in control system applications because of its simplicity in the design, analysis and implementation. However, the PID controller suffers from performance loss (mild or severe), when it is applied to higher order systems or systems with interactive dynamics. Among the second category of control systems designed using state space models are the model predictive controllers1. The model predictive controllers are widely used in industrial applications and are suitable for systems with complex dynamics and interactions between many input and output variables. Based on optimization techniques in real‐time, the model predictive controllers have the advantage of maintaining optimality for control systems in the presence of operational constraints. As a result, process efficiency and profit margins may improve. On the other hand, the deployment of online optimization techniques with operational constraints causes complexity in real‐time computation and implementation, and in an industrial environment it also leads to higher cost of maintenance and commission of such a complex control system.
When we consider the two major groups of control systems, there is an opportunity for a compromise. This compromise exists between the classical PID controllers and the modern model predictive controllers. The design formulation using a state space model for the description of system dynamics retains some of the key features of model predictive control, and avoiding online optimization solutions reduces the cost of real‐time computation, implementation and maintenance of control systems in applications. This book is to present learning materials for such a class of control systems. There are two key aspects of the associated control systems that may be attractive for practical applications. The first aspect is the design formulation using state space models, which naturally handles higher order systems or systems with interactive dynamics. The second aspect is the vector‐based control system implementation that has control signal limits imposed together with an anti‐windup mechanism, which is exceedingly simple and effective in real‐time computations.
The Book Structure
This book consists of three parts. Part I and Part II sequentially present the continuous‐time and discrete‐time state feedback control systems. Part III presents the learning materials for the Kalman filter. Because the Kalman filter has the ability to perform state estimation for linear time‐varying systems, nonlinear systems and systems with multi‐rate sampled data and missing measurement, it offers endless opportunities for extending the state feedback control systems to various challenging applications.
Part I: Continuous‐time State Feedback Control
One of the readers of this book (John) has had many decades of work experience in the field of industrial control and asked us the question why we go beyond PID control. To answer this question, the book begins by constructing a simple analytical example to show the limitations of PID control for systems with interactive dynamics. Upon completion of this detailed case study, we move on to present the fundamental theory in state feedback control (see Chapter 1), including pole‐assignment controller, linear quadratic regulator, observer and state estimate feedback control.
Disturbance rejection and reference tracking are two primary objectives for feedback control. In Chapter 2, we introduce two types of practical multivariable controllers for tracking constant reference signals and rejecting low frequency disturbances. In this chapter, we have applied the controllers to three industrial systems with design and simulation: drum boiler control, sugar mill control and drive train control of a wind turbine. MATLAB tutorials are developed in a step‐by‐step manner for the control system design and the closed‐loop simulation. We have also constructed an example to show the scenario of integrator windup and the role of anti‐windup mechanism.
Part II: Discrete‐time State Feedback Control
The continuous‐time control systems are designed in continuous‐time and are implemented in the digital environment through discretization of the control law. They are limited to systems that have a fast sampling rate. Furthermore, many complex or new systems lack a continuous‐time model from the start and face the potential complications of missing measurements in a multi‐rate sampled environment. For these regards, the discrete‐time system design and analysis offer a natural solution.
Part II of this book begins by presenting the introductory materials for discrete‐time systems (see Chapter 3). This chapter is prepared for some of us who are not familiar with the basic concepts and analysis tools in discrete‐time systems.
In parallel with the state feedback control in the continuous‐time, Chapter 4 presents the theory for discrete‐time state feedback control, including discrete‐time observer, discrete‐time linear quadratic regulator (DLQR) and DLQR with a prescribed degree of stability.
In discrete‐time, the reference signals and disturbance signals can be conveniently described by a discrete‐time model. According to the internal model principle (Francis and Wonham (1976)), the desired properties for reference tracking and disturbance rejection of a particular signal can be achieved in a stable closed‐loop control system if the signal generator is embedded into the control system. Based on these observations and the internal model principle, in Chapters 5 and 6 we develop the learning materials for discrete‐time control system design, simulation and implementation where a disturbance model is used to describe either the reference signal or the disturbance signal. In the design framework, the external signals may be commonly encountered such as step or ramp signals, or a complex signal, and the control system design and implementation remain the same. This generalization is especially useful because the tasks for reference tracking and disturbance rejection of complex signals in control system design and simulation are simplified.
Among these two chapters, Chapter 5 introduces the idea of disturbance observer and the associated control system design to compensate the effects of disturbance via observer design. In contrast, Chapter 6 is centered around the controller design for reference tracking and disturbance rejection. Repetitive control systems, which will track a periodic reference signal, are a natural extension of both control systems. For both chapters, MATLAB tutorials are developed in a step‐by‐step manner for the control system design and closed‐loop simulation. The examples include three applications: a quadruple tank, a heating furnace and a robotic arm with complex moving trajectories.
Part III: Kalman Filtering
The Kalman filter has always been fundamental in the control engineering field for the past six decades. Its importance has increased significantly in the recent decades as the Kalman filter is widely used in the navigation and localization of autonomous aerial and ground vehicles, and in the even wider applications of robotics and the modern manufacturing industry. Understanding the Kalman filter and having the capability to design and implement it are desirable and even necessary for an engineer. In Part III of this book we present the Kalman filter and the approaches associated with the implementation of the Kalman filter in a low‐cost computational device.
In the literature, there are many ways to derive the Kalman filter. In Chapter 7, we choose to derive the Kalman filter using an intuitive approach from a control engineer's point of view. Also, from this point of view, we address the commonly encountered issues such as sensor bias, load disturbance, multi‐rate sampled data and missing measurement data in state estimation. In this chapter, we give a detailed presentation on the extended Kalman filter for state estimation of nonlinear systems. Additionally, MATLAB tutorials are given in a step‐by‐step manner to show the computations of the Kalman filter, the Kalman filter with multi‐rate sampled data and the extended Kalman filter.
Considering the challenges of implementing the Kalman filter in a low‐cost computational device, Chapter 8 discusses the sequential Kalman filter to avoid matrix inversion and the Kalman filter with factorization for improving the filter's numerical accuracy when it is implemented for real‐time applications. MATLAB tutorials are given to show the implementation of the sequential Kalman filter and the Kalman filter with factorization.
Key Features of this Book
Systems and control engineering underpins many engineering disciplines. It becomes increasingly important for an engineer to have the skills of control system design, simulation and implementation. The goal of this book is to facilitate the learning activities of undergraduate and graduate students, industrial researchers and engineers in any engineering field. To achieve this goal, we combine the control system theory with various applications together with MATLAB/Simulink tutorials.
Learning by examples.
There are 68 examples and 9 major case studies in the book. The major case studies are associated with practical applications. The examples and case studies contain the illustration of the theory, numerical solutions, and closed‐loop simulations.
Learning by practice.
There are 23 MATLAB/Simulink tutorials given to show the design, simulation and implementation of the control systems and the Kalman filter. These tutorials are presented in a step‐by‐step manner so that the learner can follow the process to create their own solutions and closed‐loop simulations. Once having understood the closed‐loop simulations, we can convert the MATLAB/Simulink simulation programs into C‐code for real‐time control system implementation.
For each section, there is a set of questions for us to consider. At the end of each chapter, there is a set of problems for us to practice the design and simulation of the control systems.
The Audience
For the past 10 years, we have developed the book materials for classroom teaching of an undergraduate/graduate course across the entire engineering school in RMIT University, Australia. We have also had the privilege to teach short and intensive graduate courses based on the book materials in several other universities overseas with the delivery mode either in‐person or through the internet. The courses are typically designed using the combination of lectures and MATLAB tutorials with exercises. The students in all the classes have reported excellent learning outcomes and positive experiences after taking these control system subjects.
An undergraduate course can be developed based on the book materials from Part I in continuous‐time control systems. An advanced undergraduate or a graduate course can be developed entirely based on the book materials from Part II and Part III.
We hope that the undergraduate and graduate students in any engineering field will find this book particularly helpful due to a good amount of theory presented in a combination with applications. Because the book is practical and application oriented, we hope that the industrial researchers and engineers will find this book useful for their control system design, simulation and implementation. Finally, this book is written in a way that is enjoyable to read.
Melbourne, Australia
Liuping Wang Robin Ping Guan
Note
1
Model predictive controllers are also designed using transfer function models and step response models
Acknowledgments
We would like to express our gratitude to Professor Graham Goodwin at the University of Newcastle in Australia and Professor Robin Evans at the University of Melbourne in Australia for their help and support over the years. We wish to thank Professor Eric Rogers and Professor Chris Freeman at the University of Southampton, UK, for the collaborative work on repetitive predictive control systems. For discussions, we wish to thank Professor Kevin Moore at the Colorado School of Mines in USA, Professor Tore Hagglund at the Lund Institute of Technology in Sweden, Professor Alex Penlidis at the University of Waterloo in Canada. We wish to thank Dr Arash Vahidnia at RMIT University in Australia for discussions on the topic of drive train control of a wind‐turbine.
For valuable comments towards improvement of this book, we wish to thank Professor Antonio Visioli at the University of Brescia in Italy, Dr Minh Tran and Mr Chuong Nguyen in RMIT Vietnam, Dr John Tsing, who had worked in Measurex Corp. USA as a process control engineer and was an adjunct professor at San Jose State University, USA. We wish to thank Dr Junaid Saeed at RMIT University for drawing the block diagrams of drum boiler, quadruple tank and hot air balloon.
List of Symbols and Acronyms
Symbols
State matrix of state‐space model
Input‐to‐state matrix of state‐space model
State‐to‐output matrix of state‐space model
Laplace transfer function of controller
‐transfer function of controller
Disturbance model
Closed‐loop polynomial
Desired closed‐loop polynomial
Sampling interval
Laplace transform for output disturbance
Laplace transform for input disturbance
Laplace transform for measurement noise
Transfer function model
Performance index for optimization
Performance index for optimization with finite horizon
Feedback control gain vector, also Kalman filter gain
Observer gain vector
Condition number of
matrix
Backward shift operator,
Forward shift operator,
Pair of weighting matrices in the cost function of linear quadratic regulator, also the covariance matrices for the process noise and measurement noise in the Kalman filter
Reference signal
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MATLAB files
Lecture Slides
Part IContinuous‐time State Feedback Control
1State Feedback Controller and Observer Design
1.1 Introduction
State feedback control systems open up a different landscape to control system design for complex systems that have a higher order or have many input and output variables. For those of us who are familiar with the classical control systems designed using a Laplace transfer function model, the continuous‐time control systems using a state space model will be a natural extension. Therefore, it is easier to learn and master the design steps in the continuous‐time.
The chapter begins with a case study on the effectiveness of PID control of a system with interactions (see Section 1.2). This is to answer the question why to go beyond PID control because this question is urgent for some of us who work in the field of process control and use PID control as bread and butter for daily operations. In Section 1.3, the basic idea about state feedback control is illustrated using a simple analytical example and is followed by the introduction of controllability with an explanation what it means when a system loses its controllability. The concept of closed‐loop eigenvalues (or poles) is fundamental in state feedback control. By choosing a set of desired closed‐loop eigenvalues, a state feedback controller is designed (see Section 1.4). With a similarity transformation, this section takes us in a step‐by‐step manner for the design of a pole‐assignment controller. However, the pole‐assignment controllers have limitations when the systems have multiple inputs as illustrated in Section 1.5. For a control system with multiple inputs, the Linear Quadratic Regulator (LQR) offers a new effective design tool in which optimization underpins the design methodology (see Section 1.5). State estimation using an observer has always been an integrated part of state feedback control because it solves the fundamental problem of how to implement a control system when the state variables are not measured. Section 1.6 introduces observer design and implementation for state feedback control. It also explains the concept of observability with a simple analytical example and duality between observer and controller. The final section of this chapter (see Section 1.7) introduces the state estimate feedback control system, which is to combine the state feedback control system with the observer. Additionally, the closed‐loop stability of such a system is ensured by the stability of the state feedback control system and the observer error system.
1.2 Motivation for Going Beyond PID Control
PID control has been instrumental for the success of wide control engineering applications. Without doubt, the majority of control engineering applications used PID controllers. Before we begin our journey in studying state space control system design methods, we need to answer the question why we want to go beyond PID control. Because the PID controllers have a limited complexity, it is not difficult to imagine that when a system has complex dynamics, a controller beyond a PID structure could provide performance improvement. However, it is harder to imagine that if a system has multi‐input and multi‐output, even though it has a first order or a second order structure for the subsystems, a controller beyond PID structures could also provide performance improvement, as long as there are interactions between its inputs and outputs. For this purpose, we will explore the limitations of PID control systems in the context for systems with multiple input and output. When using PID controllers for a multi‐input and multi‐output system, the control system is completely decentralized. For those who wish to further study decentralized control systems, we refer the books by Skogestad and Postlethwaite [2007] and Goodwin et al. [2000].
The limitations of PID control system will be examined using a very simple example with respect to two aspects: (i) closed‐loop stability and dynamic performance, and (ii) steady‐state performance under the assumption that the closed‐loop system is stable.
System under study
We consider a system with two inputs and two outputs described by the following transfer function:
where () and are constants. From this transfer function model, there are interactions between the inputs and outputs. The interactions are reflected by the following relationships:
Clearly when and , the input signal will affect the output , and the same is true for the effect of output by the input signal .
Simple integral controllers
Because PID controllers are designed for single input and single output systems, in their design, we will neglect the interactions shown in (1.2) and (1.3) by regarding the constants . This is the essential assumption in the design of PID controllers, where the interactions are regarded as a form of disturbance. Thus, we will consider the following two independent systems:
These are two first order systems for which we can use two PI controllers to achieve a desired closed‐loop performance (see for example, Wang [2020]). In order to illustrate the basic ideas and keep the solution simple, we will consider two integral controllers and , where
The closed‐loop characteristic equations for the two systems are
which lead to the closed‐loop poles as
Therefore, as long as we choose and , the two closed‐loop systems will be stable with poles strictly on the left‐half of the complex plane. Note that the stability statement is true if there is no interaction in the system, meaning that .
Effect of interactions
Now, we explore what happens when there is an interaction between the two subsystems where and . When the two integral controllers and are used for the original system (1.1), we obtain the following matrix fraction models:
Figure 1.1 illustrates the closed‐loop control system in terms of reference signal (), input disturbance signal (), output disturbance signal () and measurement noise ().
Figure 1.1 Closed‐loop control system.
From this figure, the output response to the reference signal is governed by
where is identity matrix with the dimension , is called complementary sensitivity function. The output response to the output disturbance is governed by
where is called sensitivity function. The output response to the input disturbance is governed by
where is called input disturbance sensitivity function.
The next step in the investigation of the control system is to compute the closed‐loop transfer functions for reference tracking and disturbance rejection when we use the two integral controllers for the multi‐input and multi‐output system.
By using (1.10) and (1.11), we can verify that
where , and we have used the matrix inversion equality:
Similarly, we can verify that the sensitivity function
and the input disturbance sensitivity function
Upon obtaining the closed‐loop transfer functions, closed‐loop stability and performance can be analyzed. There are two topics specifically concerned in the analysis: (i) closed‐loop stability and dynamic performance, and (ii) steady‐state performance for reference tracking and disturbance rejection under the assumption that the closed‐loop system is stable.
(i) Closed‐loop stability and dynamic performance
From the closed‐loop transfer functions (1.15)–(1.18) that govern the output responses to the reference, output disturbance, and input disturbance, we see that the closed‐loop poles for the system with interactions are determined by the solutions of the characteristic equation:
There are four poles for this system. We have two cases listed as below.
Case A. If either or , then the closed‐loop poles consist of the two sets of the poles for the two single‐input and single‐output systems, which are
Therefore, as long as we choose and , the closed‐loop system is guaranteed to be stable with poles strictly on the left‐half of the complex plane. This is the ideal case.
Figure 1.2 The variations of closed‐loop poles for .
Case B. If and , then the closed‐loop poles are no longer the same as those we originally chose in Case A. The closed‐loop poles are unknown if the parameters and are unknown. In fact, the closed‐loop poles are dependent on the values of , , , and . In order to guarantee the closed‐loop system to be stable, a necessary condition for this system is that the constant term
so that the closed‐loop poles obtained through the closed‐loop characteristic equation (1.19) are strictly on the left‐half of the complex plane. The second necessary condition is not obvious, but can be determined through Routh‐Hurwitz stability criterion (see Goodwin et al. [2000], Wang [2020]), which is
If both (1.20) and (1.21) are satisfied for the parameters and with the given integral controller gains and , then the closed‐loop control system is stable. We choose to investigate the variations of closed‐loop poles as a function of . Figure 1.2 (a) shows the variations of the closed‐loop poles for . It is seen that when , the closed‐loop poles are at (two poles) and . As increases, the pair of real poles split along the real axis: one towards and the other towards . The closed‐loop pole, which moves towards as increases, will cause closed‐loop instability. Figure 1.2 (b) shows the variations of the closed‐loop poles for . It is seen that the pair of closed‐loop poles at split to become a pair of complex poles, but they are confined to the left‐half plane. In contrast, the pair of complex poles at move towards right‐half plane as increases. The necessary condition (1.21) is eventually violated as increases. By comparing Figure 1.2 (a) with Figure 1.2 (b), it seems that for the choice of , when and have an opposite sign, the closed‐loop system can tolerate a larger change in the parameters and .
In summary, when there are interactions in the system, single‐loop PID controllers could not ensure closed‐loop stability. Even if the closed‐loop system is stable, the existence of interaction will cause uncertainty in the closed‐loop dynamic response due to the changes of the closed‐loop poles from what we have chosen in the original design.
(ii) Steady‐state performance
Assume that the closed‐loop system is stable, namely, the closed‐loop poles obtained from the solutions of the characteristic equation (1.19) are strictly on the left‐half of the complex plane. Under this assumption, to analyze closed‐loop performance at steady‐state, we will apply final value theorem. Assuming that the continuous time signal has a Laplace transform , final value theorem states that
if has all poles strictly on the left‐half of the complex plane.
In the steady‐state performance analysis, we will consider both reference signal and disturbance signal to be step signals.
A. Tracking step reference signals. We assume that both reference signals to the two outputs are unit step signals so that
From (1.12) and (1.15), we obtain
With the assumption that the closed‐loop system is stable with all poles strictly on the left‐half of the complex plane, from the application of final value theorem, the final values of the output signals become:
where , the complementary sensitivity function at steady‐state, is given by
