PID Control System Design and Automatic Tuning using MATLAB/Simulink E-Book

Liuping Wang

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley - IEEE
• Sprache: Englisch

Seitenzahl: 553

Veröffentlichungsjahr: 2020

Beschreibung

Covers PID control systems from the very basics to the advanced topics This book covers the design, implementation and automatic tuning of PID control systems with operational constraints. It provides students, researchers, and industrial practitioners with everything they need to know about PID control systems--from classical tuning rules and model-based design to constraints, automatic tuning, cascade control, and gain scheduled control. PID Control System Design and Automatic Tuning using MATLAB/Simulink introduces PID control system structures, sensitivity analysis, PID control design, implementation with constraints, disturbance observer-based PID control, gain scheduled PID control systems, cascade PID control systems, PID control design for complex systems, automatic tuning and applications of PID control to unmanned aerial vehicles. It also presents resonant control systems relevant to many engineering applications. The implementation of PID control and resonant control highlights how to deal with operational constraints. Provides unique coverage of PID Control of unmanned aerial vehicles (UAVs), including mathematical models of multi-rotor UAVs, control strategies of UAVs, and automatic tuning of PID controllers for UAVs * Provides detailed descriptions of automatic tuning of PID control systems, including relay feedback control systems, frequency response estimation, Monte-Carlo simulation studies, PID controller design using frequency domain information, and MATLAB/Simulink simulation and implementation programs for automatic tuning * Includes 15 MATLAB/Simulink tutorials, in a step-by-step manner, to illustrate the design, simulation, implementation and automatic tuning of PID control systems * Assists lecturers, teaching assistants, students, and other readers to learn PID control with constraints and apply the control theory to various areas. * Accompanying website includes lecture slides and MATLAB/ Simulink programs PID Control System Design and Automatic Tuning using MATLAB/Simulink is intended for undergraduate electrical, chemical, mechanical, and aerospace engineering students, and will greatly benefit postgraduate students, researchers, and industrial personnel who work with control systems and their applications.

Leseprobe

Table of Contents

Cover

Preface

Acknowledgment

List of Symbols and Acronyms

About the Companion Website

1 Basics of PID Control

1.1 Introduction

1.2 PID Controller Structure

1.3 Classical Tuning Rules for PID Controllers

1.4 Model Based PID Controller Tuning Rules

1.5 Examples for Evaluations of the Tuning Rules

1.6 Summary

1.7 Further Reading

Problems

Notes

2 Closed-loop Performance and Stability

2.1 Introduction

2.2 Routh–Hurwitz Stability Criterion

2.3 Nyquist Stability Criterion

2.4 Control System Structures and Sensitivity Functions

2.5 Reference Following and Disturbance Rejection

2.6 Disturbance Rejection and Noise Attenuation

2.7 Robust Stability and Robust Performance

2.8 Summary

2.9 Further Reading

Problems

3 Model-Based PID and Resonant Controller Design

3.1 Introduction

3.2 PI Controller Design

3.3 Model Based Design for PID Controllers

3.4 Resonant Controller Design

3.5 Feedforward Control

3.6 Summary

3.7 Further Reading

Problems

Notes

4 Implementation of PID Controllers

4.1 Introduction

4.2 Scenario of a PID Controller at work

4.3 PID Controller Implementation using the Position Form

4.4 PID Controller Implementation using the Velocity Form

4.5 Anti-windup Implementation using the Position Form

4.6 Anti-windup Mechanisms in the Velocity Form

4.7 Tutorial on PID Anti-windup Implementation

4.8 Dealing with Other Implementation Issues

4.9 Summary

4.10 Further Reading

Problems

5 Disturbance Observer- Based PID and Resonant Controller

5.1 Introduction

5.2 Disturbance observer-Based PI Controller

5.3 Disturbance observer-Based PID Controller

5.4 Disturbance observer-Based Resonant Controller

5.5 Multi-frequency Resonant Controller

5.6 Summary

5.7 Further Reading

Problems

6 PID Control of Nonlinear Systems

6.1 Introduction

6.2 Linearization of the Nonlinear Model

6.3 Case Study: Ball and Plate Balancing System

6.4 Gain Scheduled PID Control Systems

6.5 Summary

6.6 Further Reading

Problems

7 Cascade PID Control Systems

7.1 Introduction

7.2 Design of a Cascade PID Control System

7.3 Cascade Control System for Input Disturbance Rejection

7.4 Cascade Control System for Actuator Nonlinearities

7.5 Summary

7.6 Further Reading

Problems

8 PID Controller Design for Complex Systems

8.1 Introduction

8.2 PI Controller Design via Gain and Phase Margins

8.3 PID Controller Design using Two Frequency Points

8.4 PID Controller Design for Integrating Systems

8.5 Summary

8.6 Further Reading

Problems

9 Automatic Tuning of PID Controllers

9.1 Introduction

9.2 Relay Feedback Control

9.3 Estimation of Frequency Response using the Fast Fourier Transform (FFT)

9.4 Estimation of Frequency Response Using the frequency sampling filter (FSF)

9.5 Monte-Carlo Simulation Studies

9.6 Auto-tuner Design for Stable Plant

9.7 Auto-tuner Design for a Plant with an Integrator

9.8 Summary

9.9 Further Reading

Problems

Note

10 PID Control of Multi-rotor Unmanned Aerial Vehicles

10.1 Introduction

10.2 Multi-rotor Dynamics

10.3 Cascade Attitude Control of Multi-rotor UAVs

10.4 Automatic Tuning of Attitude Control Systems

10.5 Summary

10.6 Further Reading

Problems

Suggestions to Food for Thought Questions

Bibliography

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1 Ziegler–Nichols tuning rule using oscillation testing data.

Table 1.2 Ziegler-Nichols tuning rules with a reaction curve.

Table 1.3 Cohen–Coon tuning rules with a reaction curve.

Table 1.4 Padula and Visioli tuning rules (PI controller).

Table 1.5 Padula and Visioli tuning rules (PID controller).

Table 1.6 Wang–Cluett tuning rules with reaction curve (

).

Table 1.7 PI controller parameters with reaction curve.

Table 1.8 PI controller parameters with reaction curve.

Table 1.9 PI controller parameters with reaction curve.

Chapter 2

Table 2.1 Routh–Hurwitz table.

Table 2.2 Routh–Hurwitz table for the third order system.

Table 2.3 Modified PI controller parameters with gain margin, phase margin, a...

Chapter 8

Table 8.1 Normalized PID controller parameters (

,

).

Table 8.2 Normalized PID controller parameters (

,

).

Chapter 9

Table 9.1 PID controller parameters for different

values.

Table 9.2 PID controller parameters and mean squared errors from the control ...

Chapter 10

Table 10.1 Quadrotor hardware list.

Table 10.2 Quadrotor parameters.

Table 10.3 Flight controller and avionic components.

Table 10.4 Physical specifications of the hexacopter.

List of Illustrations

Chapter 1

Figure 1.1 Proportional feedback control system.

Figure 1.2 Closed-loop step response of a proportional control system (Examp...

Figure 1.3 Proportional plus derivative feedback control system (

).

Figure 1.4 PD controller structure in implementation.

Figure 1.5 PI control system.

Figure 1.6 Closed-loop step response of a PI control system (Example 1.2).

Figure 1.7 IP controller structure.

Figure 1.8 Closed-loop step response of PI control system (Example 1.3). Key...

Figure 1.9 PID controller structure.

Figure 1.10 IPD controller structure.

Figure 1.11 Step responses of PID control system (Example 1.4). (a) Response...

Figure 1.12 Sustained closed-loop oscillation (control signal).

Figure 1.13 Comparison of closed-loop output response using Ziegler–Nichols ...

Figure 1.14 Step response data. (a) Input signal. (b) Output signal. Key: li...

Figure 1.15 Unit step response (Example 1.6)

Figure 1.16 Closed-loop unit step response with PI controller (Example 1.6)....

Figure 1.17 Unit step response (Example 1.7).

Figure 1.18 Closed-loop unit step response with PI controller (Example 1.7)....

Figure 1.19 Comparison of closed-loop responses using Padula and Visioli PI ...

Figure 1.20 Comparison of closed-loop responses using Padula and Visioli PID...

Figure 1.21 Unit step response of the fired heater process.

Figure 1.22 Comparison of closed-loop responses using Ziegler–Nichols and Wa...

Chapter 2

Figure 2.1 Closed-loop control system in transfer function form.

Figure 2.2 Nyquist plot with a unit circle for illustration of gain margin a...

Figure 2.3 Magnitude of

(solid line) together with dashed line to determin...

Figure 2.4 Nyquist plots with a unit circle (Example 2.3). Key: line (1)

u...

Figure 2.5 Nyquist plots for the modified controller with a unit circle (Exa...

Figure 2.6 Magnitude of

(solid line) together with dashed line to determin...

Figure 2.7 Comparison of closed-loop step responses (Example 2.3). Key: line...

Figure 2.8 One degree of freedom control system structure.

Figure 2.9 Two degrees of freedom control system structure.

Figure 2.10 Two degrees of freedom PI control system structure, where

and

Figure 2.11 Complementary sensitivity function with bandwidth illustration. ...

Figure 2.12 Nyquist diagrams using Padula and Visioli PID controller (Exampl...

Figure 2.13 Complementary sensitivity function using the Padula and Visioli ...

Figure 2.14 Sensitivity function using the Padula and Visioli PID controller...

Figure 2.15 Comparison of closed-loop responses using Padula and Visioli PID...

Figure 2.16 Nyquist diagrams using the Padula and Visioli PID controller (Ex...

Figure 2.17 Sensitivity functions using Padula and Visioli PID controller (E...

Figure 2.18 Closed-loop responses to disturbance and measurement noise using...

Figure 2.19 Closed-loop responses to disturbance and measurement noise using...

Figure 2.20 Unit step response of the eighth reactor with lines to assist ob...

Figure 2.21 Magnitude of modeling errors with the first order plus delay mod...

Figure 2.22 Complementary sensitivity function and graphic presentation of r...

Figure 2.23 Closed-loop step responses (Example 2.7). (a) Control. (b) Outpu...

Chapter 3

Figure 3.1 Step response of the desired closed-loop transfer function. (a)

Figure 3.2 Closed-loop response (Example 3.1). (a) Control signal. (b) Outpu...

Figure 3.3 Closed-loop response of PI control system (Example 3.2). (a) Cont...

Figure 3.4 Closed-loop response of PID control system (Example 3.4). (a) PID...

Figure 3.5 Closed-loop response of PID control system (Example 3.6). (a) PID...

Figure 3.6 Reference response of the PID control system (Example 3.7). Key: ...

Figure 3.7 Input disturbance rejection (Example 3.7). All controller structu...

Figure 3.8 Closed-loop response (Example 3.8). (a) Control signal. (b) Outpu...

Figure 3.9 Closed-loop response of resonant control (Example 3.9). (a) Distu...

Figure 3.10 Sinusoidal input disturbance rejection (Example 3.10).

Figure 3.11 Block diagram of the feedback and feedforward control system.

Figure 3.12 Three springs and double mass system.

Figure 3.13 Closed-loop responses for three springs and double mass system w...

Figure 3.14 Closed-loop responses for three springs and double mass system w...

Figure 3.15 Closed-loop responses with disturbance feedforward control for t...

Figure 3.16 Comparison between the closed-loop output responses when using d...

Chapter 4

Figure 4.1 Closed-loop response (Example 4.1). (a) Input signal. (b) Output ...

Figure 4.2 Closed-loop response (Example 4.2). (a) Input signal. (b) Output ...

Figure 4.3 Error signal and control signal in the integrator windup case (Ex...

Figure 4.4 PI controller (position form) with anti-windup mechanism (

repre...

Figure 4.5 Closed-loop response (Example 4.3). (a) Input signal. (b) Output ...

Figure 4.6 Closed-loop response (Example 4.4). (a) Input signal. (b) Output ...

Figure 4.7 Closed-loop response (Example 4.4). (a) Input signal. (b) Output ...

Figure 4.8 Closed-loop response (Example 4.5). (a) Input signal. (b) Output ...

Figure 4.9 Closed-loop response (Example 4.5). (a) Input signal. (b) Output ...

Chapter 5

Figure 5.1 Block diagram of the system for a disturbance observer-based PI c...

Figure 5.2 Block diagram of the control system using a disturbance observer.

Figure 5.3 Transfer function realization of the estimator based PI controlle...

Figure 5.4 Comparison of closed-loop control performance using an estimator ...

Figure 5.5 Comparison of closed-loop control performance using an estimator ...

Figure 5.6 Comparison of closed-loop control performance between the PI cont...

Figure 5.7 Transfer function realization of the disturbance observer-based P...

Figure 5.8 Closed-loop control performance using disturbance observer-based ...

Figure 5.9 Transfer function realization of a resonant controller with satur...

Figure 5.10 Closed-loop control response using a disturbance observer-based ...

Figure 5.11 Closed-loop control response using disturbance observer-based re...

Figure 5.12 Closed-loop control response using a disturbance observer-based ...

Figure 5.13

(Example 5.5). Key: line (1)

,

; line (2)

.

Figure 5.14 Magnitude of the sensitivity function (Example 5.6). Solid line:...

Figure 5.15 Closed-loop control response using disturbance observer-based re...

Figure 5.16 Closed-loop control response using disturbance observer-based re...

Chapter 6

Figure 6.1 Approximation of a nonlinear function at

.

Figure 6.2 Schematic of a double tank.

Figure 6.3 Schematic of the ball and plate balancing system.

Figure 6.4 Disturbance rejection. (a)

-axis response. (b)

-axis response. ...

Figure 6.5 Making a square movement. (a)

-axis response. (b)

-axis respons...

Figure 6.6 Making a circle movement. (a)

-axis response. (b)

-axis respons...

Figure 6.7 Weighting parameters.

Chapter 7

Figure 7.1 Block diagram for a system suitable for cascade control.

Figure 7.2 Block diagram of a cascade control system.

Figure 7.3 Cascade closed-loop response signals (Example 7.2, primary contro...

Figure 7.4 Closed-loop cascade control system.

Figure 7.5 Sensitivity functions for the cascade control system (Example 7.3...

Figure 7.6 Cascade closed-loop response to square wave disturbance signal wi...

Figure 7.7 One controller for disturbance rejection.

Figure 7.8 Deadzone nonlinearity.

Figure 7.9 Closed-loop control response by neglecting the actuator dynamics ...

Figure 7.10 Simulink simulation program for the cascade control system with ...

Figure 7.11 Closed-loop control response using cascade control (Example 7.5,...

Figure 7.12 Closed-loop control response using cascade control (Example 7.5,...

Figure 7.13 Illustration of a quantization of signal

with quantization int...

Figure 7.14 Closed-loop control response with quantization on the input sign...

Figure 7.15 Simulink simulation program for cascade control with actuator qu...

Figure 7.16 Cascade closed-loop control response with quantization on input ...

Figure 7.17 Illustration of a backlash nonlinearity with

and

.

Figure 7.18 The effect of backlash on closed-loop performance (Example 7.8,

Figure 7.19 The effect of backlash on closed-loop performance (Example 7.8,

Figure 7.20 Simulink simulation program for cascade control with a backlash ...

Figure 7.21 The effect of backlash on cascaded closed-loop performance (Exam...

Figure 7.22 Segment of data to illustrate the effect of backlash on cascaded...

Chapter 8

Figure 8.1 Illustration of the design process and closed-loop responses (Exa...

Figure 8.2 Closed-loop responses (Example 8.1) when using the phase margin. ...

Figure 8.3 Illustration of the design process and closed-loop responses (Exa...

Figure 8.4 Comparison of closed-loop responses of PI and PID control systems...

Figure 8.5 Comparison of closed-loop responses of PID control systems with d...

Figure 8.6 Comparison of closed-loop responses of PID control systems. (a) O...

Figure 8.7 Frequency response. Key: line (1)

; line (2)

; line (3)

Figure 8.8 Nyquist plot. Key: line (1)

; line (2)

.

Figure 8.9 Closed-loop control simulation for output stair case disturbance ...

Figure 8.10 Calculated normalized proportional controller gain. (a)

,

. (b...

Figure 8.11 Calculated gain and phase margins for PID controllers. (a) Gain ...

Figure 8.12 Calculated gain and phase margins for PI controllers. (a) Gain m...

Figure 8.13 Calculated gain and phase margins for PD controllers. (a) Gain m...

Figure 8.14 Comparison of closed-loop performance for three types of control...

Figure 8.15 Comparison of closed-loop PID control performance between the mo...

Chapter 9

Figure 9.1 Block diagram of relay feedback control.

Figure 9.2 Location of

on a Nyquist curve.

Figure 9.3 Simulink diagram for the relay feedback control.

Figure 9.4 Relay feedback control signals with hysteresis (Example 9.1). (a)...

Figure 9.5 Block diagram of integrated relay feedback control.

Figure 9.6 Location of

on a Nyquist curve when using an integrator with re...

Figure 9.7 Relay feedback control signals with and without integrator (Examp...

Figure 9.8 Frequency response estimation using FFT (Example 9.3). (a) Input ...

Figure 9.9 Frequency response estimation using FFT (Example 9.4). (a) Input ...

Figure 9.10 Block diagram of the frequency sampling filter model with reduce...

Figure 9.11 Frequency response estimation using FSF (Example 9.5).

Figure 9.12 Monte-Carlo simulation results with

random seeds in the presen...

Figure 9.13 Monte-Carlo simulation results with

random seeds in the presen...

Figure 9.14 Monte-Carlo simulation results for estimation of steady-state ga...

Figure 9.15 Simulink diagram of auto-tuner for stable system.

Figure 9.16 Nyquist plots using the PID controller parameters in Table 9.1. ...

Figure 9.17 Closed-loop simulation results using the PID controller paramete...

Figure 9.18 Closed-loop simulation results using the PID controller paramete...

Figure 9.19 Block diagram of relay feedback control for an integrating syste...

Figure 9.20 Input and output relay feedback control data (Example 9.6).

Figure 9.21 Comparison between the estimated frequency point with the origin...

Figure 9.22 Frequency response comparison (Example 9.6). (a) Nyquist diagram...

Figure 9.23 Comparison of closed-loop performance for three types of control...

Figure 9.24 Input and output relay feedback control data (Example 9.7).

Figure 9.25 Comparison between the estimated frequency point with the origin...

Figure 9.26 Comparison of closed-loop performance for three types of control...

Figure 9.27 Frequency response comparison (Example 9.7). (a) Nyquist diagram...

Figure 9.28 Relay feedback control signals from inner closed-loop system (Ex...

Figure 9.29 Auto-tuning inner-loop control system (Example 9.8). (a) Nyquist...

Figure 9.30 Relay feedback control signals from outer- closed-loop system (E...

Figure 9.31 Auto-tuning outer-loop control system (Example 9.8). (a) Nyquist...

Chapter 10

Figure 10.1 Inertial frame and body frame of the quadrotor.

Figure 10.2 Representation of a hexacopter (Ligthart et al. (2017)).

Figure 10.3 Attitude control system structure.

Figure 10.4 Quadrotor test-bed.

Figure 10.5 Experimental rig for a hexacopter.

Figure 10.6 Relay feedback control signals from the inner-loop system: top f...

Figure 10.7 Inner-loop step response in closed-loop control. Dashed line, re...

Figure 10.8 Relay feedback control signals from outer-loop system: top figur...

Figure 10.9 Comparative roll angle step response in closed-loop control.

Figure 10.10 Roll angle step response of quadrotor using test rig.

Figure 10.11 Inner loop relay test result.

Figure 10.12 Attitude control system with approximated inner loop.

Figure 10.13 Outdoor flight test.

Figure 10.14 Flight data for roll axis.

Figure 10.15 Flight data for pitch axis.

Figure 10.16 Flight data for yaw axis.

Guide

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PID Control SystemDesign and Automatic Tuning using MATLAB/Simulink

Liuping Wang

RMIT UniversityAustralia

This edition first published 2020

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Preface

PID control systems are the fundamental building blocks of classical and modern control systems. They have been used in the majority of industrial applications from chemical process control, mechanical process control, electro-mechanical process control, aerial vehicle control to electrical drive control and power converter control. Understanding these control systems and having the capability to design and implement them are paramount to a control engineer.

There are several key reasons for the continuing applications of PID controllers.

Simplicity in the design and analysis. There are three parameters to be chosen in the control systems. These parameters are easily understood and tuned by engineers.

Simplicity in the implementation. Although the PID control systems are designed and analyzed in the continuous-time, implementation is performed in discrete-time with control signal limits imposed.

The majority of the physical systems in the electrical, mechanical, aerospace and civil engineering fields can be decomposed in terms of components of first order or second order systems. For these first order and second order systems, the PID controller is a natural candidate because of its simplicity in design and implementation. For chemical process control, a complex system is often approximated using a first order plus delay model and a PID controller is commonly used.

This book is to present learning materials for students, instructors and engineers in various fields who wish to learn design, implementation and automatic tuning of PID control systems. The book begins with the basics in PID control systems (see Chapter 1), introducing the various PID control structures and the PID controller tuning rules. Chapter 2 presents the necessary tools for closed-loop stability and performance analysis and explains the roles of sensitivity functions in relation to disturbance rejection, reference following and measurement noise attenuation. In Chapter 3, pole-assignment controller design methods are introduced for PID controllers and resonant controllers that will track a sinusoidal reference signal and reject a sinusoidal disturbance. Feedforward compensation is introduced in this chapter. Many analytical examples and two MATLAB tutorials are given in this chapter to show the details of the designs. Chapter 4 discusses how a PID controller can be implemented in real-time with the topics of discretization, integrator windup problem, anti-windup mechanisms, and other implementation issues. A MATLAB real-time function is provided for PID controller implementation with anti-windup mechanism. Chapter 5 examines the PID controller design and resonant controller design from a different angle to the previous chapters. It introduces the integral mode and resonant modes through disturbance estimation. With the disturbance observer based approaches, the implementations of these control systems naturally incorporate anti-windup mechanisms when the control signal reaches its limit. MATLAB real-time functions are provided for the implementation of the PID controller and resonant controller with anti-windup mechanisms. In Chapter 6, PID control of nonlinear systems is discussed, which includes the topics of linearization, case study of a ball and plate balancing system with experimental validation, gain scheduled PID control systems and gain scheduled disturbance observer based control systems. Chapter 7 presents cascade PID control systems with the topics of cascade control system design, its roles in disturbance rejection and overcoming actuator's nonlinearities. Chapter 8 considers PID controller design for complex systems using frequency response data, which includes the topics of PID controller design using gain and phase margins, PID controller design using two frequency points with a specification on the desired sensitivity function, and empirical rules derived for PID control of integrator with time delay systems that have a performance specification and corresponding gain and phase margin measurement. MATLAB functions are given for the computation of the PID controller parameters using two frequency response points. Chapter 9 presents automatic tuning of PID controllers using relay feedback control. MATLAB real-time functions are created for relay feedback control and used for the Simulink simulations. The Fourier analysis and frequency sampling filter model are used, as two different methods, for the estimation of plant frequency response with the data generated from relay feedback control. The auto-tuners are created by linking the estimation to the PID controllers designed in the frequency domain as presented in Chapter 8. MATLAB functions are presented in a step-by-step manner for the estimation algorithms and for the auto-tuners. As case studies, Chapter 10 applies the PID control system design and the auto-tuner to multi-rotor unmanned aerial vehicles. This chapter is supported with experimental validations.

The book is self-contained with MATLAB/Simulink tutorials and supported with simulation and experimental results. Control system simulation and experimental implementation are emphasized in the book materials. The MATLAB real-time functions written for the use in Simulink simulations could be converted into C-codes for control system implementation with micro-controllers. For each section, there is a set of questions for us to reflect on. Some of them are easy and straightforward while others may require some thinking. At the end of each chapter, there is a set of problems for practicing the design and simulation of the control systems.

The book is suited for readers who have completed first three years engineering studies with some basic knowledge in block diagrams and Laplace transforms.

Liuping Wang

Melbourne, Australia

Acknowledgment

I wish to acknowledge the funding support from Mathworks Academic Support on the project entitled “PID Control Systems with Constraints: Design and Automatic Tuning using MATLAB/Simulink”. Particularly, I would like to thank Mr Bradley Horton from Mathworks for his help and support. I wish to thank Professors Shihua Li, Xisong Chen, Jun Yang and Dr Zhenhua Zhao in Southeast University, China, for interesting discussions on disturbance observer, during my visit to their university in 2014 and 2015. I wish to thank Dr Xi Chen and Dr Pakorn Poksawat previously at RMIT University Australia for their contributions on the automatic control of unmanned aerial vehicles.

For valuable comments towards improvement of this book, I wish to thank Professor Antonio Visioli at the University of Brescia, Italy, 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, Dr N. Leonard Segall of Sarnia, Ontario, Canada, Dr Chow Yin Lai, Dr Lasantha Meegahapola, Dr Arash Vahidnia, Dr Nuwantha Fernando, at RMIT University, Australia. I wish to thank Michelle Dunkley, Louis Vasanth Manoharan and Tessa Edmunds from Wiley and Sons Ltd for help and support during this book project, and Dipta Maitra for the book cover design.

I would like to thank my teaching team, Dr Robin Guan, Mr Long Tran Quang, Mr Junaid Saeed, Mr Luke McNabb and Mr Yifeng Sun, for their initiatives in teaching laboratory development. They have worked diligently to enhance students' learning experience in the subject of Advanced Control Systems at RMIT University, Australia.

List of Symbols and Acronyms

Symbols

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

Imaginary unit,

Proportional control gain

,

,

Scheduling parameters

Backward shift operator,

Sensitivity function

Input sensitivity function

Complementary sensitivity function

Desired complementary sensitivity function

Derivative control gain

Derivative control filter time constant

Integral control time constant

Closed-loop time constant

Control signal

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/Wang/PIDcontrolsystem

The website includes:

Lecture slides

MATLAB/Simulink programs

Scan this QR code to visit the companion website.

1Basics of PID Control

1.1 Introduction

This chapter will introduce the basic ideas of PID control systems. It starts with an introduction to the roles of proportional control, integral control, and derivative control, followed by an introduction to the various tuning rules developed over the past several decades. These tuning rules are mainly developed for first order plus delay systems and are simple to use. However, they do not, in general, guarantee satisfactory control system performance. Simulation examples will be used to illustrate closed-loop performance.

This chapter is suitable for those who want to understand the very basics of PID control systems. By utilization of the tuning rules, it is possible to have an application of a PID control system without further exploration.

1.2 PID Controller Structure

There are four types of controllers that belong to the family of PID controllers: the proportional (P) controller, the proportional plus integral (PI) controller, the proportional plus derivative (PD) controller and the proportional plus integral plus derivative (PID) controller. To understand the roles of the controllers, in this section we will discuss each of the structures and the PID controller parameters. From the discussions, we will establish some basic knowledge about how to use these controllers in various applications.

1.2.1 Proportional Controller

The simplest controller is the proportional controller. With this term proportional, the feedback control signal is computed in proportion to the feedback error with the formula,

(1.1)

where is the proportional gain and the feedback error is the difference between the reference signal and the output signal (). The block diagram for the closed-loop feedback control configuration is shown in Figure 1.1 where , , , and are the Laplace transforms of the reference signal, feedback error, control signal, and output signal, respectively. represents the Laplace transfer function of the plant.

Figure 1.1 Proportional feedback control system.

Because of its simplicity, the proportional controller is often used in the cases when little information about the system is available and the required control performance in steady-state operation is not demanding. As the controller only involves one parameter to be determined, it is possible to choose without detailed information about the plant.

One of the limitations of a simple proportional controller is that the steady-state error of the closed-loop control system will not be completely eliminated. We illustrate this point with the following example.

Example 1.1

Suppose that the plant is a first order system with the following transfer function,

(1.2)

with proportional controller (). Suppose that the reference signal is a step signal with amplitude 1 and its Laplace transform is . Find the steady-state value of the output with respect to the reference signal.

Solution.From Figure 1.1, the closed-loop control system from the reference signal to the plant output has the transfer function,

(1.3)

With any positive , the closed-loop system is stable where its pole is determined by the solution of the polynomial equation 1,

(1.4)

which is .

The Laplace transform of the output, , is

(1.5)

where . Applying final value theorem to the stable closed-loop system, we calculate

(1.6)

Figure 1.2 Closed-loop step response of a proportional control system (Example 1.1).

For any value of , , i.e. not equal to the desired value at the steady state response. Figure 1.2 shows the closed-loop step response with the proportional controller and , respectively. It is seen that with the increased proportional gain , the closed-loop response speed increases and the steady-state value becomes closer to the desired value 1.

1.2.2 Proportional Plus Derivative Controller

In many applications, a proportional controller is not sufficient to achieve a particular control objective such as stabilization or producing adequate damping for the closed-loop system. For instance, for a double integrator system with the transfer function:

(1.7)

the closed-loop control system with a proportional controller has a transfer function

which has a pair of closed-loop poles determined by the solutions of the polynomial equation:

These poles are at . Thus, no matter what choice we make for , the system still behaves in a sustained oscillatory manner because the pair of closed-loop poles are on the imaginary axis of the complex plane.

Now, assuming that we will additionally take the derivative of the feedback error signal into the control signal calculation, this leads to

(1.8)

where is the derivative control gain.

The Laplace transfer function of (1.8) is calculated as

Figure 1.3 Proportional plus derivative feedback control system ().

This is what we called a proportional plus derivative (PD) controller.

The closed-loop feedback control configuration for a PD controller is shown in Figure 1.3. For the double integrator system (1.7), with the derivative term included in the controller, then the closed-loop transfer function becomes

(1.9)

(1.10)

The closed-loop poles are determined by the solutions of the characteristic polynomial equation as

which are

Clearly, we can choose the values of and to achieve the desired closed-loop performance.

It is worthwhile emphasizing that almost without exception, the derivative term is different from the original form in the implementation. This is because the derivative term is not practically implementable and the differentiation of the output signal leads to amplification of the measurement noise. Thus, there are a few modifications with regard to the derivative terms. Firstly, in order to avoid the problem caused by a step reference trajectory change (the so-called derivative kick problem (Hägglund (2012))), the derivative action is only implemented on the output signal . Secondly, in order to avoid amplification of the measurement noise, a derivative filter is in a companionship with the derivative term .

A commonly used derivative filter is a first order filter and has its time constant linked as a percentage to the actual derivative gain in the form:

(1.11)

where is typically chosen to be 0.1 (10%). is chosen to be larger if the measurement noise is severe.

With the derivative filter , a typical PD controller output is calculated as

(1.12)

Figure 1.4 PD controller structure in implementation.

where is the filtered output response using (1.11). In a Laplace transform, the control signal is expressed as

(1.13)

Figure 1.4 shows the block diagram used for implementation of a PD controller with a filter.

If the derivative filter was not considered in the design, there is a certain degree of performance uncertainty due to the introduction of the filter. This may not be ideal for many applications. Designing a PD controller with the filter included will be discussed in Section 3.4.1.

1.2.3 Proportional Plus Integral Controller

A proportional plus integral (PI) controller is the most widely used controller among PID controllers. With the integral action, the steady-state error that had existed with the proportional controller alone (see Example 1.1) is completely eliminated. The output of the controller is the sum of two terms, one from the proportional function and the other from the integral action, having the form,

(1.14)

where is the error signal between the reference signal and the output , is the proportional gain, and is the integral time constant. The parameter is always positive, and its value is inversely proportional to the effect of the integral action taken by the PI controller. A smaller will result in a stronger effect of the integral action.

The Laplace transform of the controller output is

(1.15)

with being the Laplace transform of the error signal . With this, the Laplace transfer function of the PI controller is expressed as

(1.16)

Figure 1.5 shows a block diagram of the PI control system.

The example below is used to illustrate closed-loop control with a PI controller. For comparison purpose, we use the same plant as that used in Example 1.1.

Figure 1.5 PI control system.

Example 1.2

Assume that the plant is a first order system with the transfer function:

(1.17)

the PI controller has the proportional gain , and the integral time constant and 0.5 respectively. Examine the locations of the closed-loop poles. With the reference signal as a unit step signal, find the steady-state value of the closed-loop output .

Solution.We calculate the closed-loop transfer function between the reference and output signals:

(1.18)

With given in (1.16) and in (1.17), we have

(1.19)

The closed-loop poles of this system are determined by the solutions of the closed-loop characteristic equation,

(1.20)

which are

(1.21)

If the quantity

then there are two identical real poles located at

If the quantity

then there are two real poles located at

If the quantity

then there are two complex poles located at

The closed-loop system is stable as long as is positive and .

Applying the final value theorem, we calculate

(1.22)

where the steady-state value is equal to the reference signal, and it is independent of the value of integral time constant . Figure 1.6 shows for the same as in Example 1.1 the closed-loop step response with and , respectively. It is seen that as reduces, the closed-loop response speed becomes faster. Nevertheless, the steady-state responses with both values are equal to one.

It is often the case that the output of a PI control system exhibits overshoot to a step reference signal. The percentage of overshoot increases as higher control performance demanded. This may cause a conflict in the PI control performance specifications: on the one hand a fast control system response is desired, and yet on the other hand, the overshoot is not desirable when step reference changes are performed. The overshooting problem in reference change could be reduced by a small change in the configuration of the PI controller. This small change is to put the proportional control on the output signal , instead of the feedback error . More specifically, the control signal is calculated using the following relation,

(1.23)

Figure 1.6 Closed-loop step response of a PI control system (Example 1.2).

Figure 1.7 IP controller structure.

Applying a Laplace transform to this equation leads to the Laplace transform of the controller output in relation to the reference and the output as

(1.24)

Figure 1.7 shows a block diagram of this PI closed- loop control configuration. This type of implementation is called an IP controller in the literature, which is an alternative PI controller configuration. In Section 2.4, this PI controller structure is examined in the context of a reference filter within the framework of a two degrees of freedom control system. To demonstrate how this simple modification in the PI controller configuration can reduce the overshoot effect, we examine the following example.

Example 1.3

Assume that the plant is described by the transfer function:

(1.25)

and the PI controller has the parameters: , 2. Find the closed-loop transfer function between the reference signal and the output signal for the original PI controller structure (see Figure 1.5) and the IP controller structure (see Figure 1.7), and compare their closed-loop step responses.

Solution.With the PI controller in the original structure, the closed-loop transfer function between the reference signal and the output signal is calculated using,

(1.26)

By substituting the plant transfer function (1.25) and the PI controller structure (1.16), the closed-loop transfer function is

(1.27)

With the PI controller in the IP structure, the Laplace transform of the control signal is defined by (1.24). By substituting this control signal into the Laplace transform of the output via the following equation,

(1.28)

re-grouping and simplification lead to the closed-loop transfer function:

(1.29)

Figure 1.8 Closed-loop step response of PI control system (Example 1.3). Key: line (1) response from the original structure; line (2) response from the IP structure.

By comparing the closed-loop transfer function (1.27) from the original PI controller structure with the one (1.29) from the alternative structure, we notice that both transfer functions have the same denominator, however, the one from the original structure has a zero at . Because of this zero, the original closed-loop step response may have an overshoot.

Indeed, the closed-loop step responses for both structures are simulated and compared in Figure 1.8, which shows that the original PI closed-loop control system has a large overshoot; in contrast, the IP closed-loop control system has reduced this overshoot. The penalty for reducing the overshoot is the slower reference response speed.

The closed-loop transfer function obtained with IP controller structure can also be interpreted as a two degrees of freedom control system with a reference filter . This topic will be further discussed in Section 2.4.2.

Another form of PI controller, perhaps more convenient for model-based controller design as in Chapter 3, is described by:

(1.30)

This form of PI controller is identical to the original PI controller structure when the parameters of and are selected as

(1.31)

1.2.4 PID Controllers

A PID controller consists of three terms: the proportional (P) term, the integral (I) term, and the derivative (D) term. In an ideal form, the output of a PID controller is the sum of the three terms,

(1.32)