**Applied Process Control Essential Methods** Focusing on the practical implementation of the methods of process modelling and control this book provides readers with rapid access to the methods described while including the theoretical background necessary Throughout the essential knowledge is built up from chapter to chapter starting with laying the foundations in plant instrumentation and control Modelling abilities are then developed by starting from simple time loop algorithms and passing on to discrete methods Laplace transform

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Author : Michael Mulholland

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Michael Mulholland

Applied Process Control

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Michael Mulholland

Applied Process Control

Essential Methods

Author

Professor Michael Mulholland

University of KwaZulu-Natal

Chemical Engineering

4041 Durban

South Africa

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V

Essential Methods Contents

Preface XI

Acknowledgements

Abbreviations XV

Frontispiece XIX

XIII

1

1

Introduction

1.1

1.2

1.3

1.4

The Idea of Control 1

Importance of Control in Chemical Processing 3

Organisation of This Book 5

Semantics 6

References 7

2

Instrumentation

2.1

2.2

2.3

2.4

2.4.1

2.4.1.1

2.4.1.2

2.4.2

2.4.2.1

2.4.2.2

2.4.3

2.4.4

2.4.4.1

2.4.4.2

2.4.4.3

2.4.5

2.5

2.6

Piping and Instrumentation Diagram Notation 9

Plant Signal Ranges and Conversions 11

A Special Note on Differential Pressure Cells 14

Measurement Instrumentation 16

Flow Measurement 17

Flow Measurement Devices Employing Differential Pressure 17

Other Flow Measurement Devices 22

Level Measurement 22

Level Measurement by Differential Pressure 22

Other Level Measurement Techniques 25

Pressure Measurement 25

Temperature Measurement 26

Thermocouple Temperature Measurement 26

Metal Resistance Temperature Measurement 28

Temperature Measurements Using Other Principles 28

Composition Measurement 29

Current-to-Pneumatic Transducer 31

Final Control Elements (Actuators) 31

9

VI

Essential Methods Contents

2.6.1

2.6.1.1

2.6.1.2

2.6.1.3

2.6.1.4

2.6.1.5

2.6.1.6

2.6.2

2.7

2.8

2.9

Valves 32

Pneumatically Operated Globe Control Valve 32

Valve Characteristics 35

Valve CV and KV 36

Speciﬁcation of Valves for Installed Performance 37

Control Valve Hysteresis 39

Various Flow Control Devices 40

Some Other Types of Control Actuators 42

Controllers 42

Relays, Trips and Interlocks 44

Instrument Reliability 45

References 51

3

Modelling

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.9.1

3.9.2

3.9.2.1

3.9.2.2

3.9.2.3

3.9.3

3.9.3.1

3.9.3.2

3.9.3.3

3.9.3.4

3.9.3.5

3.9.4

3.9.4.1

3.9.4.2

3.9.5

3.9.5.1

3.9.5.2

3.9.6

3.9.7

3.10

3.11

General Modelling Strategy 54

Modelling of Distributed Systems 59

Modelling Example for a Lumped System: Chlorination Reservoirs 61

Modelling Example for a Distributed System: Reactor Cooler 63

Ordinary Differential Equations and System Order 67

Linearity 69

Linearisation of the Equations Describing a System 73

Simple Linearisation ‘Δ’ Concept 75

Solutions for a System Response Using Simpler Equations 77

Mathematical Solutions for a System Response in the t-Domain 77

Mathematical Solutions for a System Response in the s-Domain 79

Review of Some Laplace Transform Results 79

Use of Laplace Transforms to Find the System Response 84

Open-Loop Stability in the s-Domain 95

Mathematical Solutions for System Response in the z-Domain 97

Review of Some z-Transform Results 98

Use of z-Transforms to Find the System Response 104

Evaluation of the Matrix Exponential Terms 109

Shortcut Methods to Obtain Discrete Difference Equations 110

Open-Loop Stability in the z-Domain 111

Numerical Solution for System Response 113

Numerical Solution Using Explicit Forms 114

Numerical Solution Using Implicit Forms 115

Black Box Modelling 117

Step Response Models 117

Regressed Dynamic Models 122

Modelling with Automata, Petri Nets and Their Hybrids 126

Models Based on Fuzzy Logic 132

Use of Random Variables in Modelling 136

Modelling of Closed Loops 141

References 142

53

Essential Methods Contents

143

4

Basic Elements Used in Plant Control Schemes

4.1

4.2

4.2.1

4.2.2

4.2.3

4.2.4

4.2.5

4.2.6

4.2.6.1

4.2.6.2

4.2.6.3

4.2.7

4.2.8

4.2.8.1

4.2.8.2

4.3

4.4

4.5

4.6

4.7

4.8

Signal Filtering/Conditioning 143

Basic SISO Controllers 147

Block Diagram Representation of Control Loops 147

Proportional Controller 150

Proportional–Integral Controller 151

Proportional–Integral–Derivative Controller 153

Integral Action Windup 155

Tuning of P, PI and PID Controllers 155

Step Response Controller Tuning 158

Frequency Response Controller Tuning 159

Closed-Loop Trial-and-Error Controller Tuning 160

Feedforward Control 160

Other Simple Controllers 162

On/Off Deadband Control 162

Simple Nonlinear and Adaptive Controllers 162

Cascade Arrangement of Controllers 163

Ratio Control 164

Split Range Control 165

Control of a Calculated Variable 165

Use of High Selector or Low Selector on Measurement Signals 168

Overrides: Use of High Selector or Low Selector on Control

Action Signals 168

Clipping, Interlocks, Trips and Latching 170

Valve Position Control 171

Advanced Level Control 172

Calculation of Closed-Loop Responses: Process Model with

Control Element 173

Closed-Loop Simulation by Numerical Techniques 174

Closed-Loop Simulation Using Laplace Transforms 176

References 177

4.9

4.10

4.11

4.12

4.12.1

4.12.2

179

5

Control Strategy Design for Processing Plants

5.1

5.2

5.2.1

5.2.2

5.2.3

5.2.4

5.3

5.3.1

5.3.1.1

5.3.2

5.3.3

General Guidelines to the Speciﬁcation of an Overall Plant Control Scheme 180

Systematic Approaches to the Speciﬁcation of an Overall Plant Control Scheme 180

Structural Synthesis of the Plant Control Scheme 181

Controllability and Observability 184

Morari Resiliency Index 188

Relative Gain Array (Bristol Array) 191

Control Schemes Involving More Complex Interconnections of Basic Elements 193

Boiler Drum-Level Control 193

Note on Boiler Drum-Level Inverse Response 194

Furnace Full Metering Control with Oxygen Trim Control 195

Furnace Cross-Limiting Control 196

References 198

VII

VIII

Essential Methods Contents

199

6

Estimation of Variables and Model Parameters from Plant Data

6.1

6.1.1

6.1.2

6.1.3

6.2

Estimation of Signal Properties 199

Calculation of Cross-Correlation and Autocorrelation 199

Calculation of Frequency Spectrum 202

Calculation of Principal Components 203

Real-Time Estimation of Variables for Which a Delayed Measurement Is

Available for Correction 205

Plant Data Reconciliation 208

Recursive State Estimation 211

Discrete Kalman Filter 213

Continuous Kalman–Bucy Filter 220

Extended Kalman Filter 222

Identiﬁcation of the Parameters of a Process Model 225

Model Identiﬁcation by Least-Squares Fitting to a Batch of

Measurements 227

Model Identiﬁcation Using Recursive Least Squares on Measurements 229

Some Considerations in Model Identiﬁcation 233

Type of Model 233

Forgetting Factor 239

Steady-State Offset 240

Extraction of Physical Parameters 241

Transport Lag (Dead Time) 243

Combined State and Parameter Observation Based on a System of Differential

and Algebraic Equations 243

Nonparametric Identiﬁcation 246

Impulse Response Coefﬁcients by Cross-Correlation 246

Direct RLS Identiﬁcation of a Dynamic Matrix (Step Response) 247

References 250

6.3

6.4

6.4.1

6.4.2

6.4.3

6.5

6.5.1

6.5.2

6.5.3

6.5.3.1

6.5.3.2

6.5.3.3

6.5.3.4

6.5.3.5

6.6

6.7

6.7.1

6.7.2

251

7

Advanced Control Algorithms

7.1

7.1.1

7.1.2

Discrete z-Domain Minimal Prototype Controllers 251

Setpoint Tracking Discrete Minimal Prototype Controller 251

Setpoint Tracking and Load Disturbance Suppression with a Discrete

Minimal Prototype Controller (Two-Degree-of-Freedom Controller) 255

Continuous s-Domain MIMO Controller Decoupling Design by Inverse

Nyquist Array 256

Continuous s-Domain MIMO Controller Design Based on Characteristic Loci 259

Continuous s-Domain MIMO Controller Design Based on Largest Modulus 260

MIMO Controller Design Based on Pole Placement 261

Continuous s-Domain MIMO Controller Design Based on Pole Placement 261

Discrete z-Domain MIMO Controller Design Based on Pole Placement 264

State-Space MIMO Controller Design 266

Continuous State-Space MIMO Modal Control: Proportional Feedback 266

Discrete State-Space MIMO Modal Control: Proportional Feedback 267

Continuous State-Space MIMO Controller Design Based on ‘Controllable

System’ Pole Placement 267

7.2

7.3

7.4

7.5

7.5.1

7.5.2

7.6

7.6.1

7.6.2

7.6.3

Essential Methods Contents

7.6.4

7.6.5

7.6.6

7.7

7.7.1

7.8

7.8.1

7.8.1.1

7.8.1.2

7.8.2

7.8.2.1

7.8.2.2

7.8.2.3

7.8.2.4

7.8.2.5

7.8.3

7.8.3.1

7.8.3.2

7.8.3.3

7.8.3.4

7.8.3.5

7.8.3.6

7.9

7.9.1

7.9.2

7.10

7.11

7.11.1

7.11.2

7.12

7.12.1

7.12.2

7.13

7.14

7.14.1

7.14.2

7.14.3

7.15

Discrete State-Space MIMO Controller Design Based on ‘Controllable

System’ Pole Placement 270

Discrete State-Space MIMO Controller Design Using the Linear Quadratic

Regulator Approach 271

Continuous State-Space MIMO Controller Design Using the Linear

Quadratic Regulator Approach 277

Concept of Internal Model Control 279

A General MIMO Controller Design Approach Based on IMC 280

Predictive Control 282

Generalised Predictive Control for a Discrete z-Domain MIMO System 283

GPC for a Discrete MIMO System Represented by z-Domain Polynomials

(Input–Output Form) 284

Predictive Control for a Discrete MIMO System Represented in the

State Space 289

Dynamic Matrix Control 291

Linear Dynamic Matrix Control 296

Quadratic Dynamic Matrix Control in Industry 298

Recursive Representation of the Future Output 298

Dynamic Matrix Control of an Integrating System 300

Dynamic Matrix Control Based on a Finite Impulse Response 303

Approaches to the Optimisation of Control Action Trajectories 305

Some Concepts Used in Predictive Control Optimisation 306

Direct Multiple Shooting 309

Interior Point Method and Barrier Functions 311

Iterative Dynamic Programming 312

Forward Iterative Dynamic Programming 316

Iterative Dynamic Programming Based on a Discrete Input–Output

Model Instead of a State-Space Model 318

Control of Time-Delay Systems 320

MIMO Closed-Loop Control Using a Smith Predictor 321

Closed-Loop Control in the Presence of Variable Dead

Time 322

A Note on Adaptive Control and Gain Scheduling 323

Control Using Artiﬁcial Neural Networks 324

Back-propagation Training of an ANN 324

Process Control Arrangements Using ANNs 326

Control Based on Fuzzy Logic 328

Fuzzy Relational Model 330

Fuzzy Relational Model-Based Control 334

Predictive Control Using Evolutionary Strategies 337

Control of Hybrid Systems 341

Process Control Representation Using Hybrid Petri Nets 342

Process Control Representation Using Hybrid Automata 345

Mixed Logical Dynamical Framework in Predictive Control 350

Decentralised Control 358

References 364

IX

X

Essential Methods Contents

367

8

Stability and Quality of Control

8.1

8.2

8.2.1

8.2.1.1

8.2.1.2

8.2.1.3

8.2.1.4

8.2.1.5

8.3

8.4

8.5

8.6

8.6.1

8.6.2

8.6.3

8.6.4

8.6.5

8.7

8.8

Introduction 367

View of a Continuous SISO System in the s-Domain 369

Transfer Functions, the Characteristic Equation and Stability 369

Open-Loop Transfer Functions 369

Angles and Magnitutes of s and GO(s) 370

Open-Loop and Closed-Loop Stability 371

Open-Loop and Closed-Loop Steady-State Gain 373

Root Locus Analysis of Closed-Loop Stability 374

View of a Continuous MIMO System in the s-Domain 382

View of Continuous SISO and MIMO Systems in Linear State Space 383

View of Discrete Linear SISO and MIMO Systems 385

Frequency Response 386

Frequency Response from G(jω) 387

Closed-Loop Stability Criterion in the Frequency Domain 391

Bode Plot 393

Nyquist Plot 396

Magnitude versus Phase-Angle Plot and the Nichols Chart 401

Control Quality Criteria 403

Robust Control 404

References 408

9

Optimisation

9.1

9.2

9.3

9.4

9.5

9.5.1

9.5.2

9.5.3

9.5.3.1

9.5.3.2

9.6

9.6.1

9.7

9.8

9.8.1

9.8.2

9.9

9.9.1

9.9.2

9.9.3

9.10

9.11

Introduction 409

Aspects of Optimisation Problems 409

Linear Programming 412

Integer Programming and Mixed Integer Programming (MIP) 418

Gradient Searches 421

Newton Method for Finding a Minimum or a Maximum 421

Downhill Simplex Method 422

Methods Based on Chosen Search Directions 423

Steepest Descent Method 425

Conjugate Gradient Method 427

Nonlinear Programming and Global Optimisation 429

Global Optimisation by Branch and Bound 429

Combinatorial Optimisation by Simulated Annealing 432

Optimisation by Evolutionary Strategies 434

Reactor Design Example 435

Non-dominated Sorting Genetic Algorithm (NSGA) 437

Mixed Integer Nonlinear Programming 441

Branch and Bound Method 442

Outer Approximation Method (OA) 443

Comparison of Other Methods 444

The GAMS Modelling Environment 444

Real-Time Optimisation of Whole Plants 449

References 454

Index 457

409

XI

Preface

Material in this book is sequenced for the process engineer who needs ‘some’ background in process

control (Chapters 1–5) through to the process engineer who wishes to specialise in advanced pro

cess control (Chapters 1–9). The theory needed to properly understand and implement the methods

is presented as succinctly as possible, with extensive recourse to linear algebra, allowing multi-input,

multi-output problems to be interpreted as simply as single-input, single-output problems.

Before moving on to the more advanced algorithms, an essential practical background is laid out

on plant instrumentation and control schemes (Chapters 2, 4 and 5). Chapter 3 builds modelling

abilities from the simplest time-loop algorithm through to discrete methods, transfer functions,

automata and fuzzy logic. By the end of Chapter 5, the engineer has the means to design simple

controllers on the basis of his or her models, and to use more detailed models to test these control

lers. Moreover, ability has been developed in the use of the multi-element control schemes of

‘advanced process control’.

Chapter 6 focuses on observation. Whereas Chapter 3 reveals the tenuous chain of preparation of

plant signals, Chapter 6 aims to make sense of them. Important issues on the plant are signal con

ditioning, data reconciliation, identiﬁcation of model parameters and estimation of unmeasured

variables.

Chapter 7 addresses more advanced control algorithms, drawing on a wide range of successful

modern methods. To a large extent, continuous and discrete versions of an algorithm are presented

in parallel, usually in multi-input, multi-output formats – which simply devolve to the single-input,

single-output case if required. State–space, input–output, fuzzy, evolutionary, artiﬁcial neural

network and hybrid methods are presented. There is a strong emphasis on model predictive control

methods which have had major industrial beneﬁts.

A review of the classical methods of stability analysis is delayed until Chapter 8. This has been

kept brief, in line with reduced application in the processing industries. One recognises that stability

criteria, such as pole locations, do underlie some of the design techniques of Chapter 7. Certainly,

frequency domain concepts are part of the language of control theory, and essential for advanced

investigation. But with the slower responses and inaccurate models of processing plants, controllers

are not predesigned to ‘push the limits’ and tend to be tuned up experimentally online.

A review of a range of optimisation techniques and concepts is given in Chapter 9. Although not a

deep analysis, this imparts a basic working knowledge, enabling the development of simple applica

tions, which can then later be built upon. Topics covered include linear, integer, mixed, and non

linear programming, search techniques, global optimisation, simulated annealing, genetic algorithms

and multi-objective optimisation. These methods, and dynamic programming, underlie the

XII

Preface

predictive control and optimal scheduling topics in Chapter 7, and are also important as static opti

misers in such applications as supply chain, product blending/distribution and plant economic

optimisers.

This book tries to make the methods practically useful to the reader as quickly as possible. How

ever, there is no shortcut to reliable results, without a basic knowledge of the theory. For example,

one cannot make proper use of a Kalman ﬁlter, without understanding its mechanism. Complex

multi-input, multi-output applications will require a good theoretical understanding in order to

trace a performance problem back to a poorly calibrated input measurement. Hence, an adequate

theoretical background is provided.

A few distinctions need to be clariﬁed:

1) Modelling is a particular strength of the process engineer, and is a basis of all of the algorithms

– especially model predictive control. The reader needs to distinguish state-based models ver

sus input–output models. The state-based models can predict forward in time knowing only

the initial state and future inputs. Some algorithms rely on this. In contrast, input–output

models will need additional information about past inputs and outputs, in order to predict

future outputs. To use state-based algorithms on these, a state observer algorithm (e.g. Kal

man ﬁlter) will be required to estimate the states.

2) The forward shift operator z = eTs is used to relate discrete versions of systems to their transfer

function forms G(s) in the s (Laplace/frequency) domain. In a lot of what follows, this theoret

ical connection is not signiﬁcant, and the data sampling shift parameter q could be used, but

sometimes it is not in this text.

3) The text consistently uses bold characters to signify matrices [A], vectors [x] and matrix trans

fer functions [G(s), G(z)]. Non-bold characters are used for scalars.

A number of examples are presented in this book in order to clarify the methods. In addition, the

separate accompanying book Applied Process Control: Efﬁcient Problem Solving presents 226 solved

problems, using the methods of this text. These often make use of MATLAB code which is

arranged in obvious time loops, allowing easy translation to the real-time environment. There will,

however, be the challenge to provide additional routines such as matrix inversion.

A simple interactive simulator program has been made available at https://sourceforge.net/

projects/rtc-simulator/. It includes 20 different applications for such aspects as PID and DMC con

troller tuning, advanced level control, Smith prediction, Kalman ﬁltering and control strategies for a

furnace, a boiler and a hybrid system. No support is available for the simulator.

Although I have personally used a variety of methods on industrial and research applications, in

writing this book I have been fascinated to discover the brilliant ideas of many other workers in the

ﬁeld. To all of those people who get excited about process control, I wish you an optimal trajectory.

University of KwaZulu-Natal

March, 2016

Michael Mulholland

XIII

Acknowledgements

Many of the problems in this book are dealt with using the MATLAB program, which is distrib

uted by the MathWorks, Inc. They may be contacted at

The MathWorks, Inc.

3 Apple Hill Drive

Natick, MA 01760–2098, USA

Tel: 508-647-7000

Fax: 508-647-7001

E-mail: [email protected]

Web: mathworks.com

How to buy: http://www.mathworks.com/store

A few problems are dealt with in the GAMS optimisation environment, distributed by

GAMS Development Corporation

1217 Potomac Street, NW

Washington, DC 20007, USA

General Information and Sales: (+1) 202 342-0180

Fax: (+1) 202 342-0181

Contact: [email protected]

Some problems make use of the LPSOLVE mixed integer linear programming software which is

hosted on the SourceForge Web site at

http://sourceforge.net/projects/lpsolve/

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