Control Theory 2 - Freetwanclik.free.fr/electricity/IEPOPDF/1081ch2_1.pdf · Control Theory 2 2.1...

89

Control Theory

2

2.1CONTROL BASICS 96

Introduction 96History of Process Control 96Process Dynamics 97

Resistance-Type Processes 97Capacitance-Type Processes 98Resistance and Capacitance 98Process Gain 99Process Time Constant 99

Closing the Loop 105Oscillation and Phase Shift 105Loop Gain and Tuning 106Feedback Control 107Feedback Control Objectives 108The PID Controller 108Feedforward Control 110Feedforward Response 110Cascade Control 111Filtering 111

Advanced Controls 112References 112Bibliography 112

2.2CONTROL MODES—PID CONTROLLERS 114

Introduction 114On/Off Control 114Single-Speed Floating Control 115The Proportional Control Mode 116

Proportional Offset 116The Integral Mode 117

Reset Windup 118The Derivative Mode 118

Limitations of the Derivative Mode 119Inverse Derivative Control Mode 120

PID Algorithm Variations 121Digital Algorithms 121

Sample-and-Hold Algorithms 122Conclusions 122Bibliography 123

2.3CONTROL MODES—PID VARIATIONS 124

PID Algorithms 124Noninteracting Form 124Interacting Form 125Parallel Form 126

Set-Point Response Softening 127Derivative and Proportional Actions 127Set-Point Filtering 127

Windup Accommodation 128Reset and External Feedback 128PIDτd Algorithm 129Bibliography 129

2.4CONTROL MODES—DIGITAL PID CONTROLLERS 130

PID Position Algorithms 130PID Velocity Algorithms 132Control Structures 133Bibliography 134

© 2006 by Béla Lipták

90 Control Theory

2.5CONTROL MODES — CLOSED-LOOP RESPONSE 135

Introduction 135Linear Systems 135

Time and Frequency Domain Analysis 136Open- and Closed-Loop Control 137

Open-Loop vs. Closed-Loop Control 137Stability and Dead Time 138Advantages of Closed-Loop Control 139

Closed-Loop Transfer Functions 140Block Diagram Algebra 140Overall Transfer Functions 141

Stability and Dynamic Performance 143Following the Set Point 143Load Disturbance 144Dynamic Response 144Frequency-Function-Based Evaluation 144Loop Shaping 145

Conclusions 146Bibliography 146

2.6CONTROL SYSTEMS — CASCADE LOOPS 148

Introduction 148Cascade System Advantages 148

Components of the Cascade Loop 148The Secondary Loop 148Secondary Control Variables 149Cascade Primary Loop 150Cascade Application Examples 152Cascade Controller Design and Simulation 153

Summary 155Bibliography 156

2.7EMPIRICAL PROCESS OPTIMIZATION 157

Levels of Optimization 157Empirical Optimization 157

Optimization 157Providing Process Data 159Search Procedure 160

Conclusions 161References 161Bibliography 161

2.8EXPERT SYSTEMS 162

Artificial Intelligence 162Expert Systems 162Statistical Process Control (SPC) 165Artificial Neural Networks 165Future Trends 165Fuzzy Logic and Neural Networks 165Neural Networks 166

Summary 171References 171Bibliography 172

2.9FEEDBACK AND FEEDFORWARD CONTROL 173

Feedback Control 173Limitations of Feedback Control 173Best-Possible Feedback Control 173Integrated Error 174

Feedforward Control 175Load Balancing 175Steady-State Model 175Dynamic Model 176Adding a Feedback Loop 178Linear and Bilinear Feedforward Systems 178Performance 179Variable Parameters 179

References 180Bibliography 180

2.10GENETIC AND OTHER EVOLUTIONARY ALGORITHMS 181

Introduction 181Applications 181

The Concept of EA 181Fitness Function: Encoding the Problem 181Model of Natural Selection 182

Genetic Algorithm 182Genetic Programming 183Evolutionary Strategy 184Evolutionary Programming 184System Identification 184

Polymerization Reactor Example 185Complete Process Models 186

Process Control Applications 186Controller Tuning 186Application of IEC in Controller Tuning 186Control Structure Design 187Online Applications 187

Software Tools 188Application-Oriented Systems 188Algorithm-Oriented Systems 190Tool Kits 190


2.11HIERARCHICAL CONTROL 193

Hierarchical Levels 193History 194

The Central Computer 194Distributed Control 196


Contents of Chapter 2 91

Hierarchical Control 197Overall Tasks of Digital Control Systems 198Detailed Task Listings 200Lower-Level Computer Tasks 201Higher-Level Computer Tasks 203

Bibliography 203

2.12INTERACTION AND DECOUPLING 205

Introduction 205Interacting Process Example 205

Decoupling the Process 205Generalizing the Solution 206Measuring the Interactions 207

Merits, Drawbacks, and Compromises 207Drawbacks 207Partial and Static Decoupling 208The Reactor–Flasher Example 208


2.13MODEL-BASED CONTROL 209

The Structure of MBC 209Modeling Approaches 209Internal Model Control (IMC) 210Model Predictive Control (MPC) 211Process-Model-Based Control (PMBC) 212Summary 213Bibliography 213

2.14MODEL-BASED PREDICTIVE CONTROL PATENTS 214

Introduction 214Patent Basics 214Basic Patent History 215Online Patent Databases 215Patent Contents 216

Guide 216Forms 216Functions 218

References 222

2.15MODEL-FREE ADAPTIVE (MFA) CONTROL 224

Single-Loop MFA Control System 224MFA Controller Architecture 224SISO MFA Control Algorithm 225MFA and PID 225MFA Control System Requirements 226SISO MFA Configuration 226

Nonlinear MFA Controller 227Nonlinear MFA Configuration 227

MFA pH Controller 227MFA pH Controller Configuration 228

Feedforward MFA Controller 228Feedforward MFA Controller

Configuration 228Antidelay MFA Controller 228Robust MFA Controller 229

Robust MFA Controller Configuration 230Time-Varying MFA Controller 230

Time-Varying MFA Controller Configuration 230

Antidelay MFA pH Controller 231Multivariable MFA Control System 231Two-Input Two-Output MFA Control System 231

2 × 2 MFA Controller Configuration 231MIMO MFA Controller Application Guide 232MFA Control Methodology 232Summary 232

Simple Solution 232Use All Information Available 232Information’s Accuracy 233Technique That Fits the Application 233

References 233

2.16MODELING AND SIMULATION OF PROCESSES 234

Types of Simulations 235Steady-State Simulation 235Dynamic Simulations 235Real-Time Simulations 235

Steady-State Simulations 236Software Packages 237

Dynamic Simulations 237Temperature and Vaporization 238Pressure and Water Hammer 238pH and Solubility 239Sensors, Transmitters, and Final Control

Elements 239Simulation Languages 239

Real-Time Simulation 239Future Trends 240Virtual Plant 241References 241Bibliography 241

2.17MODEL PREDICTIVE CONTROL AND OPTIMIZATION 242

Model Predictive Control Principles 243MPC vs. Feedback Control Summary 243

Process Modeling 244Process Modeling Equations 245

Process Model Identification 245FIR and ARX Modeling 245


92 Control Theory

Model Predictive Controller 246MPC Controller Formulation 246MPC Controller Equations 247

Integrating MPC, Constraints,and Optimization 248

Optimization Equations 248MPC Optimal Controller 249MPC and Optimizer Integration 250

MPC Application Development 250Commissioning MPC Application 251

Conclusions 251Bibliography 251Abbreviations 252Definitions 252

2.18NEURAL NETWORKS FOR PROCESS MODELING 253

Neural Networks for Black-Box Modeling 253Structure of Neural Networks 254

McCulloch–Pitts Neuron 254Feedforward Multi-Layer Neural

Networks 254Example of an ANN 255

Training of Feedforward Neural Nets 255Feedforward Computation 255Weight Adaptation 256

Developing and Building Neural Networks 256Concept 257Design, Implementation and Maintenance 257Data Collection and Preparation 257Applying the Data for Validation 257Hidden Layers, Nodes, and Algorithms 258

Applications 258Identification of Dynamic Systems 258Gray-Box Modeling 259Neural Networks in Process Control 260Plant Monitoring and Fault Detection 261Intelligent Soft Sensors 262Industrial Applications 262


2.19NONLINEAR AND ADAPTIVE CONTROL 265

Introduction 265Definitions and Terminology 265

Steady-State and Dynamic Adaptation 265Approaches to Adaptive Control 266

Feedforward or Gain Scheduling 266Feedback Adaptation or Self Adaptation 269

Intelligent Adaptive Techniques 270Intelligent Identification and/or Tuning 270Multiple Model Adaptive Control

(MMAC) 271


2.20OPTIMIZING CONTROL 274

Introduction 274Defining Optimum Performance 274Finding Minimum and Maximum Points 275

Optimization Considerations 276Feedback or Feedforward Optimization 276Reducing Set Point Variability 276Evolutionary Optimization (EVOP) 277Feedforward Optimization 277

Optimizing Tools 278Linear Programming 278Nonlinear Programming 279

Constraint Handling Methods 279Gradient Search Method 280

Multivariable Noninteracting Control 280Constraint Following 281


2.21PID ALGORITHMS AND PROGRAMMING 284

PID Configurations 284Algorithms 285

Digital Algorithms 285The Position Algorithm 286The Velocity Algorithm 286Advantages and Disadvantages 287Cascade and Ratio Control 288Feedforward Control 289Other Digital Algorithms 289

Process Control Programming 290Types of Programs 290Features of Process Control Programs 291The Executive Program 291Programming Languages for Process

Control 292References 294Bibliography 295

2.22PROCESS GAINS, TIME LAGS, REACTION CURVES 296

Introduction 296Process Gains 296Dead Times and Time Constants 296

Instrumentation Effect on Process Dynamics 297

Transportation Lag 297



Dead Time Representation 298Dead Time Approximations 298

Reaction Curves 300First-Order Plus Dead Time Processes 300Underdamped Processes 302Integrating Plus Dead Time Processes 303

Conclusion 304References 304Bibliography 304

2.23RATIO CONTROL 305

Introduction 305Flow Ratio Control 305

Advantages and Disadvantages 306Ratio Stations 306Setting the Ratio Remotely 307Ratio Control Applications 307

Blending 307Surge Control of Compressors 309

Bibliography 309

2.24REAL-TIME PERFORMANCE ASSESSMENT 311

Introduction 311Plant Assessment and Monitoring 311

Loop Performance and Health Assessment 311Related Assessments Criteria 312Key Assessment Criteria 313Plant Overview 314Prioritization of Problem Areas 314Performance Monitoring Software

Capabilities 315Case Studies 317

Model Predictive Controlled Refinery 317Grade Change in a Pulp Mill 317

Conclusions 317Reference 317Bibliography 317

2.25RELATIVE GAIN CALCULATIONS 318

The Relative Gain Array 318Properties of λ 319

Calculation Methods 319The Ratio of Partial Derivatives 319Two Loops Open 320Two Loops Closed 320The Two-Slope Method 320The Matrix Method 321Reducing the Array 321

Decoupled Relative Gains 321

Partial Decoupling 321References 322Bibliography 322

2.26ROBUSTNESS: A GUIDE FOR SENSITIVITYAND STABILITY 323

Robustness Plots 323Dynamic Parameters 324Design Guidelines 324

Conclusion 325Bibliography 325

2.27SAMPLED DATA CONTROL SYSTEMS 326

Sampled Data Control Systems 326Symbols Used 326Properties of SDCS 327Mathematical Aspects of SDCS 327Design Aspects 329Dead-Beat Controller Algorithm 331Smith Predictor Design 334

Conclusions 334Bumpless Transfer and Controller Windup 334Selection of the Sampling Rate 335

Bibliography 335

2.28SELECTIVE, OVERRIDE, AND LIMIT CONTROLS 336

Introduction 336Overrides 336

Overriding at a Fixed Point 337Override to Guarantee Valve Closure 338Start-Up and Shut-Down Overrides 339

Selective Control 340Limited Availability of Manipulated

Variable 340Selecting from Multiple Transmitters 341High–Low Selection 341

Antireset Windup Overrides 341The PI Controller 341

Design Procedure 343Bibliography 344

2.29SELF-TUNING CONTROLLERS 345

Introduction 345Evolution 345

Self-Tuning Regulator (STR) 346Model-Based Methods 347


94 Control Theory

Pattern Recognition Methods 347Performance 349Conclusions 349References 349Bibliography 350

2.30SLIDING MODE CONTROL IN PROCESS INDUSTRY 351

Design Procedure 351First Step 351Second Step 351

Applications 353Electromechanical Systems 353Chemical Systems 353SMC for SISO Chemical Process Control 354SMC for MIMO Chemical Process Control 356

SMCR Implementation of PID Algorithm 357SMCr Implementation Methodology 358


2.31SOFTWARE FOR FUZZY LOGIC CONTROL 360

Introduction 360Principle of Fuzzy Systems 361

Fuzzy Sets 361Fuzzy Systems 362Direct and Supervisory Fuzzy Control 365Classical Fuzzy Control Algorithms 365

Model-Based Fuzzy Control 367Inverse Fuzzy Model-Based Control 368Fuzzy Model-Based Predictive Control 368Operating Regime-Based Modeling 368Takagi–Sugeno Fuzzy Models 369

Software and Hardware Tools 370Project Editor 371Rule Base and Membership Functions 372Analysis and Simulational Tools 372Code Generation and Communication

Links 372Conclusions 372References 373

2.32STABILITY ANALYSIS, TRANSFER FUNCTIONS 375

Introduction 375Laplace Transforms 375

Theorems 376First-Order Lag 376Partial Fraction Expansion 377

Z Transforms 379

First-Order Lag 380Inverse z Transform 380

State Space Representation 381Vector and Matrix Operations 381Second-Order Model 381Multiple-Input Multiple-Output 382

Transfer Functions 383Second-Order Lag 383PID Controllers 383Multiple-Input Multiple-Output 383

Block Diagrams 384Linearization 385Graphic Representations 387

Bode Plots 387Nyquist Plots 388

Stability 389Descartes’ Rule of Signs 390Routh’s Criterion 390Nyquist Criterion 391


2.33STATE SPACE CONTROL 393

Introduction 393State Space Description 393Control Law Design 394

State Variable Feedback 394State Feedback with PI Control 396

Observer Design 396Full Order Observer 397Full Order Observer for Constant Error 398Reduced Order Observer 398

Combined Observer–Controller 400Combined Observer-

Controller Behavior 400Transfer-Function Interpretation 402

Conclusions 403Bibliography 404

2.34STATISTICAL PROCESS CONTROL 405

Introduction 405SPC and Process Control 405Continuous Processes 406What Is Statistical Control? 406

SPC Tools and Techniques 406Control Charts 406Charts and Tables 407Interpretation of Charts 408The Purpose of the Charts 409Using Charts 409



Process Capability 410Identifying Causes 411Implementing SPC Concepts 411

Data Storage, Computation, and Display 412Bibliography 413

2.35TUNING PID CONTROLLERS 414

Disturbances 414Sources 414Dynamics 414Step Responses 415Simulating a Load Change 415Comparing Set-Point and Load Responses 416Set-Point Filtering 416

Open-Loop Tuning 417Process Reaction Curve 417Process Model 418Integral Criteria Tuning 422Which Disturbance to Tune for 422Lambda Tuning 422Digital Control Loops 424

Closed-Loop Response Methods 425Ultimate Method 425Damped Oscillation Method 427

Comparison of Closed and Open Loop 427Frequency Response Methods 427

Obtaining the Frequency Response 428PID Tuning Based on Frequency Response 428

Fine Tuning 429Optimum Load Response 429Effect of Load Dynamics 429

Symbols, Abbreviations 430Bibliography 431

2.36TUNING LEVEL CONTROL LOOPS 432

Introduction 432Idealized Model 432

Time Constant of the Tank 433Determining Tuning Parameters 434Example 436

Nonideal Processes 437Irregular Vessel Shapes 437No Cascade Loop 437Dead Time 438Unequal In- and Outflows 438Flashing Liquids 438Sinusoidal Disturbance 438

Other Approaches to Tuning 439Averaging Level Control 439Controller Gain and Resonance 440Nonlinear Gain 441

Reference 441Bibliography 441

2.37TUNING INTERACTING LOOPS, SYNCHRONIZING LOOPS 442

Introduction 442Multiloop Systems 442

Control Loop Analysis 442Interacting Loops 442

Tuning to Eliminate the Interaction 443Synchronizing Loops 444Bibliography 445

2.38TUNING BY COMPUTER 446

Introduction 446Process Modeling 447

Time and Frequency Domains 447State Space Models 448Time Series Models 448

Fitting Simple Models 449Step Response 449Modified Step Response 449Ultimate Gain Test 450Discrete Cycling 450

Tuning Criteria and Formulas 450Tuning and Diagnostic Software 452



96

2.1 Control Basics

C. F. MOORE (1970, 1985) B. G. LIPTÁK (1995)

K. A. HOO, B. G. LIPTÁK, M. J. PIOVOSO (2005)

INTRODUCTION

This section begins with some control-related definitions ofsome basic terms and concepts. This is followed by someintroductory discussion of process dynamics and closed loopfeedback control. For an in-depth discussion of analog anddigital PID (proportional-integral-derivative) variations, closedloop responses, and the various model-based expert and opti-mizing control systems, the reader should turn to the latersections of this chapter.

During the first industrial revolution, the work done byhuman muscles was gradually replaced by the power ofmachines. Process control opened the door to the second indus-trial revolution, where the routine functions of the human mindand the need for the continuous presence of human observerswere also taken care of by machines. Today, during the thirdindustrial revolution, we are designing computers that can opti-mize and protect the safety of our processes not only by self-tuning, but also by applying neural networks capable of self-learning the characteristics of the controlled process, similarto the way a baby learns about the outside world.

This way, the human operator is relieved of the bulk oftedious and repetitive physical and mental tasks and is able toconcentrate on the more creative aspects of the operating indus-try. In the third industrial revolution the traditional goal of max-imizing the quantity of production will be gradually replaced bythe goal of maximizing the quality and durability of the producedgoods, while minimizing the consumption of energy and rawmaterials and maximizing recycling and reuse. Optimized pro-cess control will be the motor of this third industrial revolution.

HISTORY OF PROCESS CONTROL

The fly-ball governor was the first known automatic controlsystem. It was installed on Watts’ steam engine over 200 yearsago in 1775. As shown in Figure 2.1a, the fly-balls detected thespeed of shaft rotation and automatically opened up the steamsupply when a drop was registered in that speed. The users ofthe fly-ball used it without understanding why it works. Anothercentury went by before James Clark Maxwell in 1868 preparedthe first mathematical analysis of the fly-ball governor.

The spreading use of steam boilers contributed to theintroduction of other automatic control systems, includingvarious steam pressure regulators and the first multiple-ele-ment boiler feed water systems. Here again the applicationof process control was ahead of its theory, as this first feed-forward control system was introduced at a time when eventhe term “feedforward” had not yet been invented, let aloneunderstood. Actually, the first general theory of automaticcontrol, written by Nyquist, did not appear until 1932.

In 1958 Donald P. Eckman’s classic Automatic ProcessControl was published, and even after that, for severaldecades, most universities, and academia as a whole, treatedthe teaching of process control as if it were just anothercourse in mathematics. Some still do and teach process con-trol as if we lived in the frequency and not in the time domain.

It was not until the late 1960s that the first edition of thishandbook and F. G. Shinskey’s classic Process Control Sys-tems started to give recognition to process control analysisin the time domain. This is no surprise as progress was alwaysmade by the practical users and not by the mathematicallyoriented theoreticians. One should also understand thatbefore one can control a process one must fully understandthe process, and for this reason this section will start withthe description of the “personalities” of different processes.

FIG. 2.1aThe fly-ball governor keeps shaft rotation constant by opening theseam supply valve when rotation slows and closing it when therotation rises.

Steamsupply


2.1 Control Basics 97

PROCESS DYNAMICS1

It is self-evident that the pilots of an ocean liner and a super-sonic airplane need to have different personalities becausethe natures and the characteristics of their vehicles are dif-ferent. This is also true in industrial process control. The pilot(the controller) must be correctly matched to the process thatit controls. In order to do that, the designer of a process mustunderstand the “personality” of the process.

Most processes contain resistance, capacitance, and dead-time elements, which determine their dynamic and steady-stateresponses to upsets. Before the control of processes is dis-cussed, these three elements of the “personalities” of processeswill be described.

To describe the personalities of processes, block diagramsare used (Figure 2.1b). The two main symbols used in anyblock diagram are a circle and a rectangular box. The circlerepresents algebraic functions such as addition or subtractionand is always entered by two lines but exited by only one.

The rectangular box always represents a dynamic function,such as a controller, where the output is the controlled variable(c) and is a function of both time and of the input or manipulatedvariable (m). Only one line may enter and only one line mayleave a rectangular block. The “system function,” the “person-ality” of the process component, is placed inside the block, andthe output is determined by product of the system function andthe input.

Resistance-Type Processes

Pressure drop through pipes and equipment is the mostobvious illustration of a resistance-type process. Figure 2.1cillustrates the operation of a capillary flow system, whereflow is linearly proportional to pressure drop. This processis described by a steady-state gain, which equals the resis-tance (R). Therefore as the input (m = flow) changes fromzero to m , the output (c = head) will go in an instantaneousstep from zero to c = Rm.

Laminar resistance to flow is analogous to electrical resis-tance to the flow of current. The unit of resistance is sec/m2

in the metric and sec/ft2 in the English system and can becalculated using Equation 2.1(1):

2.1(1)

where h = head, ft; ν = kinematic viscosity, ft2/sec; L = lengthof tube or pipe, ft; D = inside diameter of pipe, ft; q = liquidflow rate, ft3/sec; µ = absolute viscosity, lb-sec/ft2 = γ ν/g;γ = fluid density, lb/ft3.

When the flow is turbulent, the resistance is a functionof pressure drop, not of the square root of pressure drop, aswas the case with laminar (capillary) flow. A liquid flowprocess usually consists of a flow-measuring device and acontrol valve in series, with the flow (c) passing through both.Figure 2.1d illustrates such a process. As can be seen fromthe block diagram, the flow process is an algebraic and propor-tional (resistance) only process. The manipulated variable (m)is the opening of the control valve, while the controlledvariable (c) is the flow through the system. A change in mresults in an immediate and proportional change in c. Theamount of change is a function of the process gain, also calledprocess sensitivity (Ka).

The load variables of this process are the upstream anddownstream pressures (u0 and u2), which are independent,uncontrolled variables. A change in either load variable willalso result in an immediate and proportional change in thecontrolled variable (c = flow). The amount of change is afunction of their process sensitivity or gain (Kb).

FIG. 2.1bBlock diagram descriptions of algebraic and dynamic functions.

Rdhdq

L

g D= = 128

42v

πftsec/

r e m c

b

+

–

Algebraic functione = r– b

Dynamic functionc = f (m, t)

FIG. 2.1cPhysical example of a resistance element and its block diagram.

Response tounit step change

of mRm

0

0 Time, t

Out

put v

aria

ble,

c

c, Head

Capillary

Physical diagram

Block diagram

m, flow

m c

c = Rm

c = Output variable (head)R = Resistancem = Input variable (flow)

Where

R


98 Control Theory

Capacitance-Type Processes

Most processes include some form of capacitance or storagecapability. These capacitance elements can provide storage formaterials (gas, liquid, or solids) or storage for energy (thermal,chemical, etc.). Thermal capacitance is directly analogous toelectric capacitance and can be calculated by multiplying themass of the object (W) with the specific heat of the materialit is made of (Cp). The units of thermal capacitance areBTU/°F in the English or Cal/°C in the metric system.

The capacitance of a liquid or a gas storage tank canboth be expressed in area units (ft2 or m2). Figure 2.1eillustrates these processes and gives the corresponding equa-tions for calculating their capacitances. The gas capacitanceof a tank is constant and is analogous to electric capacitance.The liquid capacitance equals the cross-sectional area of thetank at the liquid surface, and if the cross-sectional area isconstant, the capacitance of the process is also constant atany head.

A tank having only an inflow connection (Figure 2.1f) is apurely capacitive process element. In such a process the level (c)will rise at a rate that is inversely proportional to the capaci-tance (cross-sectional area of the tank) and after some timewill flood the tank. The level (c) in an initially empty tank witha constant inflow can be determined by multiplying the inflowrate (m) with the time period of charging (t) and dividing thatproduct with the capacitance of the tank (c = mt/C).

Figure 2.1f illustrates such a system and describes boththe system equations and the block diagram for the capacitiveelement. In arriving at the system function the operationalnotation of the differential equation is used, using the sub-

stitution of s = d/dt. The system function therefore is 1/Csand the output (c = head) is obtained by multiplying thesystem function with the input (m).

Resistance and Capacitance

Resistance and capacitance are perhaps the most importanteffects in industrial processes involving heat transfer, mass

FIG. 2.1d The relationship between the manipulated variable (m) and thecontrolled variable (c) in a resistance (flow) process, shown bothin its physical and in its block diagram representations.

u0 u1 u2

u2 Kb

Kb

Ka+

++

–c

u0

KV

Kv is the valve sensitivity =

c

m

m

�rottling valveMetering

device

cm

FIG. 2.1e Definitions of liquid and gas capacitance.

FIG. 2.1fExample of a pure capacitance element.

q

h

Pw

C, Capacitance

C, Capacitance

Gas capacitance is defined by

Where v = Weight of gas in vessel, LB.

V = Volume of vessel, ft3

R = Gas constant for a specific gas, ft/degT = Temperature of gas, degp = Pressure, lb/ft2

n = Polytropic exponent is between 1.0 & 1.2 for uninsulated tanks

ft2Liquid capacitance is defined by C = dvdh

ft2C = =dvdp

VnRT

1Cs

dcdt

ddt

m, flow c, head m

m

c

C = m = (Cs)c = m ∴ c =

Physical diagram Block diagram

1Cs

Where C = Capacitance c = Output variable (head)

t = Time

s = = differential operatorm = Input variable (flow)



transfer, and fluid flow operations. Those parts of the processthat have the ability to store energy or mass are termed“capacities,” and those parts that resist transfer of energy ormass are termed “resistances.” The combined effect of sup-plying a capacity through a resistance is a time retardation,which is very basic to most dynamic systems found in indus-trial processes.

Consider, for example, the water heater system shown inFigure 2.1g, where the capacitance and resistance terms can bereadily identified. The capacitance is the ability of the tank andof the water in the tank to store heat energy. A second capaci-tance is the ability of the steam coil and its contents to storeheat energy. A resistance can be identified with the transfer ofenergy from the steam coil to the water due to the insulatingeffect of a stagnant layer of water surrounding the coil.

If an instantaneous change is made in the steam flow rate,the temperature of the hot water will also change, but thechange will not be instantaneous. It will be sluggish, requir-ing a finite period of time to reach a new equilibrium. Thebehavior of the system during this transition period willdepend on the amount of material that must be heated in thecoil and in the tank (determined by the capacitance) and on

the rate at which heat can be transferred to the water (deter-mined by the resistance).

Process Gain

The system function is the symbolic representation of how aprocess component changes its output in response to a changein its input. For example, if the system function is only again, only a constant (Kc, G, or some other symbol) wouldappear inside the block. The process gain is the ratio betweenthe change in the output (dc) and the change in the input thatcaused it (dm). If the input (m) to the block is a sinusoidal,the output (c) will also be a sinusoidal.

The process gain can be the product of a steady-stategain (Kp) and a dynamic gain (gp) component. If the gainvaries with the period of the input (exciting) sinusoidal, itis called dynamic gain (gp), while if it is unaffected by thisperiod it is called steady-state gain (Kp). Therefore, if theprocess gain can be separated into steady-state and dynamiccomponents, the system function can be given inside theblock as (Kp)(gp).

The dynamic gain (gp) appears as a vector having a scalarcomponent Gp and a phase angle. It will be shown shortlythat capacitance-type process elements do introduce phaseshifts in such a way that the peak amplitude of an inputsinusoidal does not cause a simultaneous peak amplitude inthe output sinusoidal.

Process Time Constant

Combining a capacitance-type process element (tank) witha resistance-type process component (valve) results in asingle time-constant process. If the tank was initially emptyand then an inflow was started at a constant rate of m, thelevel in the tank would rise as shown in Figure 2.1h and

FIG. 2.1g A water heater has both resistance and capacitance effects.

Steam

Cool water

Hot waterTank

Drain

Steamcoil

FIG. 2.1h A single time-constant process consists of a capacitance and a resistance element. The time it takes for the controlled variable (c) to reach63.2% of its new steady-state value is the value of the time constant.

Response of output

(c)

Out

put v

aria

ble,

c

Time, t/T

Unit step-change (A)in input flow (Mo)

at time = 0

R Mo = B

0.632 R Mo

T

A c, head m, flow

Physical diagram

R q 4 3 2 1 0

m R Ts + 1

Block diagram

�e steady-state gain of the process is

Kp = B/A

c


100 Control Theory

would eventually rise to the steady-state height of c = Rmin the tank.

In order to develop the overall system function, one hasto combine the capacitance element of Figure 2.1f with aresistance element. If the input variable is the inflow (m) andthe output variable is the level (c), the tank capacitance equalsthe difference between inflow (m) and outflow (q):

2.1(2)

In the fluid resistance portion of the system, the output (thehead c) equals the product of the outflow (q) and the resistance(R). Because c = qR, therefore q = c/R. Substituting c/R for qin Equation 2.1(2) and multiplying both sides by R gives

2.1(3)

The unit of R is time divided by area and the unit of Cis area; therefore, the product RC has the unit of time. Thistime (T) is called the time constant of the process. It has beenfound experimentally that after one time constant the valueof the output variable (c) of a single time-constant processwill reach 63.2% of its final value.

Process elements of this description are common and aregenerally referred to as first-order lags. The response of a first-order system is characterized by two constants: a time constantT and a gain K. The gain is related to the amplification asso-ciated with the process and has no effect on the time charac-teristics of the response. The time characteristics are relatedentirely to the time constant. The time constant is a measureof the time necessary for the component or system to adjustto an input, and it may be characterized in terms of the capac-itance and resistance (or conductance) of the process.

In their responses (Figure 2.1h), two characteristics dis-tinguish first-order systems:

1. The maximum rate of change of the output occursimmediately following the step input. (Note also that ifthe initial rate were unchanged the system would reachthe final value in a period of time equal to the timeconstant of the system.)

2. The actual response obtained, when the time lapse isequal to the time constant of the system, is 63.2% ofthe total response.

These two characteristics are common to all first-order processes.

Characteristic Equations of First-Order Systems A first-ordersystem is one in which the system response to a constantchange in the input is an exponential approach to a new con-stant value. Such a system can be described by a first-orderdifferential equation:

2.1(4)

where u(t) and y(t) are the deviations of the inputs and out-puts, respectively, and a and b are constants. The response

of such a system with a = 1 and b = 2 to a unit step change(value 1 for t ≥ 0 and 0 for t < 0) in the input is shown inthe top panel of Figure 2.1i.

This result can be demonstrated from the solution of thedifferential equation in Equation 2.1(4). The solution of thisdifferential equation is important because it provides insightinto the operation of the system and nature of the controlproblem.

The solution of any linear differential equation can bebroken into the sum of two parts: the homogeneous and theparticular solution. The homogeneous solution is that expres-sion that when substituted for y(t) in the left-hand side ofEquation 2.1(4) produces the value zero. The particular orsteady-state solution is that part of the total solution that isstrictly due to the forcing input, u(t).

To establish the homogeneous solution, assume the solu-tion to be of the form

2.1(5)

with u(t) = 0 in Equation 2.1(4). Substituting Equation 2.1(5)into the left-hand side of Equation 2.1(4) imposes the fol-lowing condition on p:

2.1(6)

which is satisfied with . It is clear from Equation2.1(5) that for a < 0, the homogeneous solution becomesunbounded. The total solution, which is the sum of the homo-geneous and particular solutions, becomes dominated by thehomogeneous solution and not the particular solution.

Systems in which the homogeneous solutions grow with-out bound are said to be unstable. The homogeneous solutiondoes not depend on the form of the forcing input and henceit is a characteristic of the system. In fact, a polynomial

C dc dt m q( / ) = −

RC dc dt c Rm( / ) + =

dy t

dtay t bu t

( )( ) ( )+ =

FIG. 2.1i Step response of a first-order system with and without dead time.

Step response of a first-order system with a 1 minute dead time

Step response of a first-order system2

1

00 1 2 3 4 5 6

0 1Time in minutes

2 3 4 5 6

1.5

0.5Resp

onse

y(t)

2

1

0

1.5

0.5Resp

onse

y(t)

tc

y t Kehpt( ) =

p a+ = 0

p a= −



equation such as Equation 2.1(6) that defines yh(t) is calledthe characteristic equation.

The particular solution, , is that part of the totalsolution that is due entirely to the forcing function, u(t). Toestablish yp(t), assume that it has the same functional formas u(t), provided that the homogeneous solution does not havethe forcing function as part of it. For example, if u(t) is aramp, assume that yp(t) is also a ramp with unknown interceptand slope. If yh(t) contains a term that is a ramp, then assumethat the particular solution is quadratic in time with unknowncoefficients. Substitute the assumed form into Equation2.1(4) and solve for the unknowns.

EXAMPLE

As an example, suppose that u(t) in Equation 2.1(4) is aunit step. Assuming that the system is stable, the outputwill eventually reach a constant. Therefore, the particularsolution is assumed to be a constant, α . Substituting αfor y(t) and 1 for u(t) in Equation 2.1(4), yields that α =b/a. Note that in Figure 2.1i, the particular solution is 2,obtained with b = 2 and a = 1.

The speed of response of a first-order system is deter-mined by the time constant, tc. The time constant is the timeat which the exponent in the homogenous solution is −1.From Equation 2.1(5) and Equation 2.1(6), this correspondsto 1/a. Figure 2.1i illustrates the time constant and output forthe first-order system given in Equation 2.1(4).

If tc is small, then the particular solution decays rapidlyand the steady-state solution is quickly reached. Such a sys-tem is said to be “fast.” On the other hand, if tc is large, thenthe system reaches the steady state very slowly and is saidto be “slow” or “sluggish.”

First-Order System with Dead Time A first-order system cou-pled with dead time, or process delay, makes an excellent modelfor many process systems. The bottom panel of Figure 2.1iillustrates a first-order system with a 1-min dead time. Further-more, this model is not difficult to determine. A simple steptest in which the input to the system is changed from one valueto another while the system is at some steady-state output givessufficient information to determine the model.

Suppose the input is changed by an amount ∆. Let theoutput reach its new steady-state value, and divide the dif-ference between the new and old steady-state outputs by ∆.This computed response is called the step response. The deadtime is the time at which the output first starts to respond tothe change in the input. If the exponent in the solution isevaluated at the time constant, the exponent of the particularpart would be −1. This corresponds to the decay of 63.2%or of the initial contribution of the particular solution.Thus, the time constant can be determined as the differencebetween the time at which the response is at 63.2% of thefinal value and the dead time.

In Figure 2.1i, 63.2% of the final response, denoted bythe horizontal dashed line, has a value of 1.264 and yields a

time constant of one. Experience shows that first-order plusdead time systems can be controlled by proportional plusintegral (PI) controllers.

Multiple Time-Constant Processes Figure 2.1j illustrates aprocess where two tanks are connected in series and thereforethe system has two time constants operating in series. Theresponse curve of such a process (r = 2 in Figure 2.1k) is

y tp ( )

1 1− −e

FIG. 2.1jThe physical equipment and the block diagram of a process withtwo time constants in series.

FIG. 2.1kThe responses of processes slow as the number of their (equal size)time constants in series increases.

m u2

u1

R1

R1u1

u2T1s + 1

c

cm+

+++

h2

R2

R1 R2T1s + 1

R2

h2

T2s + 1

R2T2s + 1

1.0

0.8

0.6

0.4

0.2

00 1

r = 1

2

3

4

6

2Time, t/T

m c

Out

put v

aria

ble,

c

3 4 5

1(Ts + 1)r


102 Control Theory

slower than that of the single time-constant process (r = 1 inFigure 2.1k) because the initial response is retarded by thesecond time-constant. Figure 2.1k illustrates the responses ofprocesses having up to six time constants in series. As thenumber of time constants increases, the response curvesbecome slower (the process gain is reduced), and the overallresponse gradually changes into an S-shaped reaction curve.

Characteristic Equations of Second-Order Systems Second-order systems are characterized by second-order differentialequations. These systems are more complex than first-orderones. A second-order system might be viewed as the solutionof the following differential equation:

2.1(7)

Again, y(t) is the process output (controlled variable) andu(t) is the input, which is usually the reference input function.The parameters and in this equation dictate the natureof the solution. The quantity is called the damping ratioand is the natural frequency. If is zero, then the solu-tion is a sinusoid that oscillates at ω n radians per second(Figure 2.1l). Thus, the system is an oscillator producing asinusoidal response at the natural frequency of the oscillator.

The damping ratio is the ratio of the damping constantto the natural frequency. The damping constant determinesthe rate at which the oscillations decay away. As the dampingconstant increases, the speed of response increases and theoscillations decay more rapidly.

Figure 2.1m illustrates the effect of the damping ratio onthe response of a second-order system. The top left panel of

Figure 2.1m corresponds to the case of the damping ratiobeing zero. Note that the oscillations will continue foreverand the steady-state solution is embedded in the response.This is an undamped response.

The top right panel corresponds to 0 < ξ < 1 and producesan underdamped output. The speed of the response has dete-riorated slightly in that the output y(t) is a little slower ingetting to the desired output 1. Note that the response oscil-lates, but the oscillation decays leaving only the steady-statesolution.

The bottom left panel corresponds to being criticallydamped (ξ = 1). The response does not exceed the steady-state value of 1, but the speed of the system is noticeablyslower than either the undamped or underdamped cases. Thebottom right panel is the overdamped case (ξ > 1). Again, theresponse system is much slower.

From a control perspective, if the response is allowed toexceed the target value the system response is faster due tothe underdamped nature of the system behavior. In fact, manyheuristic tuning methods seek parameters that produce anunderdamped response provided that the overshoot, or theamount that the output exceeds its steady-state response, isnot too large.

If the second-order system also has dead time associatedwith it, then its response is not unlike that of the first-ordersystem. That is, the response is delayed by the dead time, butthe overall approach to steady state for different values ofthe damping ratio remains the same. A first-order systemcontrolled with a PI controller may produce underdampedsecond-order system behavior.

Dead-Time Processes A contributing factor to the dynam-ics of many processes involving the movement of mass fromone point to another is the transportation lag or dead time.

FIG. 2.1lIllustration of the effect of damping ratio variations from 0 to 5 ona second-order system.

d y t

dt

dy tdt

y t u tn n n

2

22 22

( ) ( )( ) ( )+ + =ξω ω ω

ξ ωn

ξωn ξ

c(t)/K

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0 2 4 6wn t

8 10 12

x = 0

x = 0.3

x = 0.7

x=3.0x=5.0

x = 0.1

x=1.5

x=1.0

FIG. 2.1mThe effect of the damping ratio on second-order systems.

2

1

1

0

00 5 10

0 5 10 0 5 10

10

1.5

1.5

0.5

1

00 5

0.5

0.80.6

0.4

0.2

1

0

0.80.6

0.4

0.2

Critically damped

Undamped Underdamped

Overdamped



Consider the effect of piping on the hot water to reach alocation some distance away from the heater (Figure 2.1n).

The effect of a change in steam rate on the water tem-perature at the end of the pipe will not depend only on theresistance and capacitance effects in the tank. It will also beinfluenced by the length of time necessary for the water tobe transported through the pipe. All lags associated with theheater system will be seen at the end of the pipe, but theywill be delayed. The length of this delay is called the trans-portation lag (L) or dead time. The magnitude is determinedas the distance over which the material is transported (l)divided by the velocity at which the material travels (v). Inthe heater example

2.1(8)

Dead time is the worst enemy of good control, and the processcontrol engineer should therefore make a concentrated effortto minimize it. The effect of dead time can be compared todriving a car (the process) with closed eyes or with thesteering wheel disconnected during that period. The goal ofgood control system design should be both to minimize theamount of dead time and to minimize the ratio of dead timeto time constant (L/ T).

The higher this ratio, the less likely it is that the controlsystem will work properly, and once the L/T ratio reaches1.0 (L = T ), control by traditional PID (proportional–integral–derivative) strategies is unlikely to work at all. Thevarious means of reducing dead time are usually related toreducing transportation lags. This can be achieved by increas-ing the rates of pumping or agitation, reducing the distancebetween the measuring instrument and the process, eliminat-ing sampling systems, and the like.

When the nature of the process is such that the L/T ratiomust exceed unity, or if the controlled process is inherently adead-time process (a belt feeder for example), the traditional

PID control must be replaced by control based on periodicadjustments, called sample-and-hold type control (see thenext sections of this chapter).

Process Variables Many external and internal conditionsaffect the performance of a processing unit. These conditionsmay be detected in terms of process variables such as temper-ature, pressure, flow, concentration, weight, level, etc. Mostprocesses are controlled by measuring one of the variables thatrepresent the state of the process and then by automaticallyadjusting another variable(s) that determine that state. Typi-cally, the variable chosen to represent the state of the systemis termed the “controlled variable” and the variable chosen tocontrol the system’s state is termed the “manipulated variable.”

The manipulated variable can be any process variable thatcauses a reasonably fast response and is fairly easy to manip-ulate. The controlled variable should be the variable that bestrepresents the desired state of the system. Consider the watercooler shown in Figure 2.1o. The purpose of the cooler is tomaintain a supply of water at a constant temperature. Thevariable that best represents this objective is the temperatureof the exit water, Two, and it should be selected as the controlledvariable.

In other cases, direct control of the variable that bestrepresents the desired condition is not possible. Consider thechemical reactor shown in Figure 2.1p. The variable that is

FIG. 2.1n Transportation lag introduces a dead time in the water heater process.

Hot watervelocity

TZ

Tl

L = distance

Water

Steam

L v l= /

FIG. 2.1o Process variables in a simple water cooler.

FIG. 2.1pProcess variables in a simple chemical reactor.

Coolant inlet:

Coolant outlet:

Water outlet:

Water inlet:

Wc = Flow rateTci = Temperature

Ww = Flow rateTwo= Temperature

W = Flow rateTwi = Temperature

Wc = Flow rateTco = Temperature

Steam

Reactants:

Reactor:

Product:

Wr = Flow rate

Steam condensate

Wp = Flow rate

PR = Reactor pressure (PSIG)TR = Reactor temperature (°F)Cor = Composition

Ci = Composition

Cop = Composition

Qs = Flow rateHs = Enthalpy


104 Control Theory

directly related to the desired state of the product is thecomposition of the product inside the reactor. However, inthis case a direct measurement of product composition is notalways possible.

If product composition cannot be measured, some otherprocess variable is used that is related to composition. Alogical choice for this chemical reactor might be to hold thepressure constant and use reactor temperature as an indicationof composition. Such a scheme is often used in the indirectcontrol of composition.

Degrees of Freedom To fully understand the “personality”of a process, one must also know the number of variablesthat can be independently controlled in a process. The max-imum number of independently acting automatic controllersthat can be placed on a process is the degrees of freedom(df ) of that process. Mathematically, the number of degreesof freedom is defined as

2.1(9)

where df = number of degrees of freedom of a system;v = number of variables that describe the system; and e =number of independent relationships that exist among thevarious variables.

It is easy to see intuitively that a train has only one degreeof freedom because only its speed can be varied, while boatshave two and airplanes have three (see Figure 2.1q).

The degrees of freedom of industrial processes are morecomplex and cannot always be determined intuitively. In the caseof a liquid-to-liquid heat exchanger, for example (Figure 2.1r),the total number of variables is six, while the number of definingequations is only one—the first law of thermodynamics, whichstates the conservation of energy.

Therefore, the degrees of freedom of this process are 6 −1 = 5. This means that if five variables are held constant, this

will result in a constant state for the sixth variable, the outlettemperature (c). Therefore, the maximum number of auto-matic controllers that can be placed on this process is five.Usually one would not exercise this option of using fivecontrollers, and in the case of a heat exchanger of this type,one might use only one control loop.

One would select the controlled variable (c) to be theprocess property that is the most important, because it hasthe most impact on plant productivity, safety, or productquality. One would select the manipulated variable (m) to bethat process input variable that has the most direct influenceon the controlled variable (c), which in this case is the flowrate of the heating fluid. The other load variables (u1 to u4)are uncontrolled independent variables, which, when theychange, will upset the control system, and their effects canonly be corrected in a “feedback” manner.

This means that a change in load variables is notresponded to until it has upset the controlled variable (c). Inorder to fully describe this process, one would also considersuch system parameters as the mass or the specific heat ofthe liquids, but such parameters are not considered to bevariables.

When the process involves a phase change, the calcula-tion of the degrees of freedom follows Gibbs’s phase rule,stated in Equation 2.1(10):

2.1(10)

where n = number of chemical degrees of freedom; nc =number of components; and np = number of phases.

For example, if the process is a boiler producing satu-rated steam (Figure 2.1s), the number of components is one(H2O), the number of phases is two (water and steam), and,therefore, the number of degrees of freedom is n = 1 − 2 +2 = 1. Consequently, only one variable can be controlled:temperature or pressure, but not both. If a boiler produces

FIG. 2.1qThe degrees of freedom of an airplane.

df v e= −

Altitude

N

EW

S

Variables:

Equations:

Degree of freedom:None

Altitude 1Latitude 1Longitude 1

3

3–0

FIG. 2.1r The degrees of freedom of a liquid-to-liquid heat exchanger.

m

6 Variables:

c = heated fluid outlet temperaturem = heating fluid flow rateu1 = heating fluid inlet temperatureu2 = heating fluid outlet temperatureu3 = heated fluid flow rateu4 = heated fluid inlet temperature

u2

u4 c, u3

u1

n n nc p= − + 2



superheated steam, the number of degrees of freedom is two,and therefore both temperature and pressure can be indepen-dently controlled.

When the process is more complex, such as is the caseof binary distillation, the calculation of the degrees of free-dom also becomes more involved. Figure 2.1t lists 14 vari-ables of this process, not all are independent. Since there aretwo components and two phases at the bottom, feed andoverhead, Gibbs’s law states that only two of the three variables(pressure, temperature, and composition) are independent.Therefore, the number of independent variables is only 11. Thenumber of defining equations is two (the conservation ofmass and energy), and therefore, the number of degrees offreedom for this process is 11 − 2 = 9. Consequently, notmore than nine automatic controllers can be placed on thisprocess.

CLOSING THE LOOP2

After gaining a good understanding of the “personality” ofthe process and after having identified the controlled (c),manipulated (m), and uncontrolled load (u) variables, one canproceed to “close the loop” by installing an automatic con-troller onto the process.

The task of a controller is to measure the controlledvariable c (say the level, in the case of Figure 2.1u), comparethat measurement to some desired target value (r = set point),and, if there is an error (c does not equal r), modify its outputsignal (m) and thereby change the manipulated variable m(input to the process) in a way that will reduce the deviationfrom the set point.

Oscillation and Phase Shift

When a closed loop is in sustained oscillation, the period andamplitude of oscillation are constant, and therefore, the totalphase shift in the loop must be 360 degrees. This state iscalled sustained oscillation or marginal stability.

In order to keep a ball bouncing at the same period andamplitude, it has to be struck at phase intervals of 360 degrees,when it is at its highest point. If it is struck sooner, the periodof oscillation will be shortened. If the ball is the process andthe hand is the controller, the goal is to push the ball backtoward set-point whenever it is moving away from it.

FIG. 2.1sA saturated steam boiler has only one degree of freedom.

FIG. 2.1tIn a binary distillation process the number of independent variablesis 11 and the number of degrees of freedom is nine.

Q = Heat input (BTU/hr)

Saturatedvapor

Water

Ws = Steam flow

Ps = Pressure

Feed

Steam(V )

L Overhead product

Bottom product(B)

(D)

Apparentvariables:

Independentvariables:

C1 = Overhead temperatureC2 = Overhead pressure 2

2

2

C3 = Overhead compositionC4 = Overhead flow rate 1

1

111

11

U1 = Bottom temperatureU2 = Bottom pressureU3 = Bottom compositionU4 = Bottom flow rateU5 = Feed temperatureU6 = Feed pressureU7 = Feed compositionU8 = Feed percent vaporU9 = Feed flow ratem = Steam flow rate (heat input)

FIG. 2.1u The addition of a controller closes the automatic control loop.

Manipulatedvariable (m)

Manipulated

variable (m)

Controlled variable(c)

Controlledvariable

(c)

Devia

tion

(e =

r – b

= r –

c)

Uncontrolledload variable (u)

Set point(r)

R

m

LC


106 Control Theory

Controlling a process is different from bouncing a ballonly in that the force applied by the controller is appliedcontinuously and not in periodic pulses. On the other hand,the two processes are similar from the perspective that in orderto sustain oscillation the application of these forces must bedisplaced by 360 degrees (Figure 2.1v). The sources of thisphase lag can be the process, the controller dynamics (modes),and the negative feedback of the loop.

The response of a process to a sinusoidal input is alsosinusoidal if the process can be described by linear differen-tial equations (a linear process). The phase shift in suchprocesses is usually due to the presence of capacitance ele-ments. For example, in the case of the process shown inFigure 2.1w, the process causes the output (c = outflow fromthe tank) to lag behind and oscillate with a smaller amplitudethan the input of the process (m = inflow to the tank).

Loop Gain and Tuning

It was shown in Figure 2.1v that sustained oscillation resultsif the loop gain is 1.0, and quarter amplitude damping resultsif the loop gain is 0.5. The goal of the tuning of most processcontrol loops is to obtain quarter amplitude damping. This willresult if the product of all the gains in the loop comes to 0.5.This end result is achieved through tuning, which is the processof finding the controller gain, which will make the overall gainproduct 0.5.

The controller gain (and most other gains also) consistsof a steady-state component (Kc = the proportional setting),which is unaffected by the period of oscillation and thedynamic gain (gc), which varies with the period of the input(error) sinusoidal. The (Kc)(gc) product is the total gain ofthe controller (Gc). Therefore, slow processes can be con-trolled with high-gain controllers, while controllers on fastprocesses must have low gains.

An example of a slow (low-gain) process is space heating,where it takes a long time (accumulation of the heat inputprovided by manipulated variable) to cause a small change

in the controlled variable (the room temperature). If the com-bined gain of that process is 0.02, the controller gain requiredto provide quarter amplitude damping is 25 (0.02 × 25 = 0.5).

If, on the other hand, the controlled process is a flowprocess, or a pH process near neutrality, the process gain ishigh. In such cases the controller gain must be low so thatthe total loop gain is 0.5.

The higher the controller gain, the more corrective actionthe controller can apply to the process in response to a smalldeviation from set point and the better the quality of theresulting control will be. Unfortunately, if the gain product

FIG. 2.1v Control loops respond to upsets differently as a function of the loop gain and of the phase lag contributions of both the process and thecontroller.

l0A1

A2

A2 = 1/4 A1

Loop responsesfollowing anupset:

Unstablelooprun-awayoscillation

Stable looptuned for 1/4amplitudedamping

Stage ofmarginalstability

Loop gain: >1.0

Variable –180� <180�

1.0 0.5

Dynamic phase lagcontributed by thecontroller and theprocess:

ˆ

l0̂

FIG. 2.1wThe outflow lags behind the inflow in this process because fluidmust accumulate in the tank before the level will rise. This accu-mulation in turn will increase the outflow due to the resulting higherhydrostatic head.

m

m

m

Process phase lag

Sign

al am

plitu

de

c

c

c

C

R

1Ts + 1



of controller and process reaches unity, the process becomesunstable and undamped oscillations (cycling) will occur.Therefore, it is not possible to tightly control fast (high-gain)processes without cycling. It is easier to obtain tight controlon slow, low-gain processes, because the use of high-gaincontrollers does not destroy stability.

The loop gain is the product of all the gains in the loop,including sensor, controller, control valve, and process. In aproperly tuned loop, the product of all these gains is 0.5.What makes tuning difficult is that the process gain oftenvaries with process load. For example, in heat transfer pro-cesses, when the heat load is low and the heat transfer surfaceavailable to transfer the heat is large, the transfer of heat isperformed efficiently; therefore, under those conditions, thisprocess is a high-gain process.

As the load rises, the same heat transfer process becomesa low-gain process because the fixed heat transfer areabecomes less and less sufficient to transfer the heat. There-fore, as shown in Figure 2.1x, the gain of a heat transferprocess (Gp) drops as the load rises.

Tuning such a system can be a problem because in orderto arrive at an overall loop gain of 0.5, the controller shouldapply a high gain when the load is high and a low gain whenthe load is low. Standard controllers cannot do that becausethey have been tuned to provide a single gain. Therefore, ifthe loop was tuned (controller gain was selected) at high loads,

the loop will cycle when the load drops, and if the loop wastuned at low loads, the loop will not be able to hold the processon set point (will be sluggish) when the load rises.

One way to compensate for this effect is to install acontrol valve in the loop (an equal percentage control valve),which increases its gain as the load rises. When the gain ofthe process drops, the gain of the valve increases and thetotal loop gain remains relatively unaffected.

Feedback Control

Two concepts provide the basis for most automatic controlstrategies: feedback (closed-loop) control and feedforward(open-loop) control. Feedback control is the more commonlyused technique of the two and is the underlying concept onwhich most automatic control theory used to be based. Feed-back control maintains a desired process condition by mea-suring that condition, comparing the measurement with thedesired condition, and initiating corrective action based onthe difference between the desired and the actual conditions.

The feedback strategy is very similar to the actions of ahuman operator attempting to control a process manually.Consider the control of a direct contact hot water heater. Theoperator would read the temperature indicator in the hot waterline and compare its value with the temperature desired(Figure 2.1y). If the temperature was too high, he wouldreduce the steam flow, and if the temperature was too low,he would increase it. Using this strategy, he would manipulatethe steam valve until the error is eliminated.

An automatic feedback control system would operate inmuch the same manner. The temperature of the hot water ismeasured and a signal is fed back to a device that comparesthe measured temperature with the desired temperature. If anerror exists, a signal is generated to change the valve positionin such a manner that the error is eliminated.

The only real distinction between the manual and auto-matic means of controlling the heater is that the automaticcontroller is more accurate and consistent and is not as likelyto become tired or distracted. Otherwise, both systems con-tain the essential elements of a feedback control loop.

FIG. 2.1xIf the process gain varies with load, such as is the case of a heattransfer process, the gain product of the loop can be held constantby using a valve whose gain variation with load compensates forthe process gain variation.

Load

Valve gain(Gv)

Gp

Gv

Gs

Gc

m1

e

bc

m

Gc � Gv � Gp � Gs = 0.5

Sensorgain(Gs)

Setpoint (r)

Cont

rolle

rga

in(G

c)

Processgain(G

p)

Load

Load

Load

Load(u)

+ +

+ –

For stablecontrol

FIG. 2.1yManually accomplished feedback control.

Waterheater

Temperatureindicator

Hot water

Desiredtemperature = To

Steam

Cool waterSteamvalve


108 Control Theory

Feedback control has definite advantages over othertechniques in relative simplicity and potentially successfuloperation in the face of unknown contingencies. In general,it works well as a regulator to maintain a desired operatingpoint by compensating for various disturbances that affectthe system, and it works equally well as a servo system toinitiate and follow changes demanded in the operating point.

Feedback Control Objectives

First, it is desirable that the output follow the desired behav-ior. Note that in all the subplots in Figure 2.1m, the responseto a unit step change approaches the steady-state value of 1,which corresponds to the magnitude of the input stepresponse. Because the output response eventually reaches thesteady-state value, the steady-state error, or the differencebetween the desired final output and the actual one, is zero.

Second, almost always, the steady-state error should bezero to a step input, or constant targets, as inputs. In somecases, such as the case of a ramp input, it also is desirable forthe steady-state error to be zero or nearly so. There may be anupper limit on the magnitude that is tolerable when no distur-bances are present. However, in the presence of disturbancesthe steady-state error can become larger.

Third, the speed of response is important. From the dis-cussion in connection with equation 2.1(7), viz. the solutionof the differential equation, the steady-state is attained as thehomogeneous portion of the solution of the differential equa-tion approaches zero. A control system can affect the rate atwhich this happens. If the response of the system is sluggish,then the output (control action) of the controller is not changingenough in magnitude in response to the difference between thedesired and actual output. By changing the parameters of thecontroller, the magnitude of the control action and the speedof response can be increased in response to control errors.

Fourth, the physical limitations of the plant constrain theability of the controller to respond to input commandchanges. Another measure of the controller’s speed is thesettling time. The settling time is defined as the time afterwhich the control system will remain within a given percentageof the desired final value when there are no outside distur-bances. Figure 2.1z illustrates a “2% settling time,” meaningthe time it takes for a step response to approach the finalsteady-state value within 2%.

Lastly, note that (Figure 2.1l) the step change responsesof a second-order system all have an overshoot, when thedamping ratio of the system is less than one. Overshoot isdefined as the percentage by which the peak response valueexceeds the steady-state value (peak value of step response −steady-state value)/(steady-state value). A small overshoot canbe acceptable, but large overshoots are not.

The PID Controller

In Sections 2.2, 2.3, and 2.4 detailed descriptions are pro-vided of the various analog and digital PID algorithms andtherefore only some of the basic aspects are discussed here.

As was shown earlier, the dynamic behavior of manyprocesses can be modeled adequately as first- or second-ordersystems. Thus, the PID is an ideal tool with which to controlsuch systems. Furthermore, the PID is easily understood andtuned by trained operators.

Consider the feedback control system with a plain pro-portional-only controller shown in Figure 2.1aa. Assume thatthe process to be controlled is a static system with a gain Kp.For a proportional-only controller, the controller output is theproduct of the error signal (e = r − c) and of the proportionalgain Kc. That is,

2.1(11)

The closed-loop response is the relationship between theoutput (controlled variable), c, and the reference or set pointinput, r. This relationship is

2.1(12)

Note that if r is a constant, say one, the controlled output isless than one. Thus, there is a nonzero steady-state error toconstant inputs. This is not surprising because if r = c, thene = r − c = 0 and the output of the controller would also be

FIG. 2.1z The step response shown has a 4.2-second “2% settling time.”

FIG. 2.1aa Block diagram of a PID feedback control loop.

Step response

Time (sec)

1.4

1.2

1

0.8

0.6

0.4

0.2

00 1 2 3 4 5 6

Am

plitu

de

2% settling time

u K ec=

cK K

K Krc p

c p

=+1

r e cu+

−PID Plant



zero. This produces a contradiction in that c would be forcedto zero, which, in general, is not the value of r.

Therefore, the plain proportional controller reduces butdoes not eliminate the error. Note from Equation 2.1(12) thatas the controller gain increases, the controlled output, c,approaches the referenced input r more and more closely.Thus, the steady-state error, e, becomes smaller as Kc is madelarger. But since Kc

can never be infinite, the error is neverzero.

The Derivative Mode To better understand the effect ofderivative action, consider the situation in which the control-ler shown in Figure 2.1aa is proportional plus derivative (PD).In this case, the controller output is given by

2.1(13)

if the proportional gain Kc is 1 and the derivative gain is ,which is called the derivative time. Figure 2.1bb illustratesthe controlled variable’s step response, the error and deriva-tive of the error signal, when the set point (reference) is r = 1.

The vertical lines in the top two plots of Figure 2.1bbare in locations where the controlled variable signal and theerror signal have local maximums and minimums or pointswhere the derivative is zero. Note that the controlled variableresponse has exceeded the set point (target value) of one. Theexcess overshoot is due to the presence of a certain momen-tum in the response of the system; the controller did not turn

the input around in time to stop the system from exceedingthe desired value.

This is seen in the error signal that has not changed signuntil the controlled variable output has exceeded the set point.Note also that the derivative is zero at the peak values andof opposite sign to the value of the error signal. When theerror is added to a constant times the derivative, the result isa signal that changes sign earlier in time, that is, before theoutput has actually reached the steady-state value. Figure 2.1ccillustrates this.

The PD controller is used in applications where overshootcannot be tolerated, such as pH neutralization. The reductionor elimination of overshoot can often be accomplished with-out significantly affecting the settling time of the closed-loopsystem. The primary disadvantage of derivative mode is itssensitivity to noise. Noise is generally of high frequency, anddifferentiating just amplifies it. The controller output canbecome cyclic or unstable, which can have a detrimentaleffect on the longevity of actuators such as valves or motors.

Integral Mode Nearly all controllers have some form of inte-gral action. Integral action is important because it correctsbased on the accumulated error, which is the area under theerror curve. If the error goes to zero, the output of the integratoris the constant area that had accumulated up to that point.

Consider the feedback system illustrated in Figure 2.1aa.The task of the integral term in the PID algorithm is to findthe manipulated variable (the input to the plant) needed todrive the steady-state error to zero when the set point (refer-ence input) is constant.

When the error is zero, both the proportional term andthe derivative term contribute nothing to the controller output.

FIG. 2.1bbThe step response of a PD controller showing the responses of thecontrolled variable (top), the error (center), and the derivative ofthe error (bottom).

u t e t Tde t

dtd( ) ( )( )= +

Td

1.5

1

0.5

00 1 2 3 4 5

Error signal

Step response

6 7 8 9 10

1

210

0.5

0

−0.5

−1−2

0 1 2 3 4 5Derivative of error signal

6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

FIG. 2.1cc The “anticipation” of the PD controller can be seen by noting thecontrolled variable response of a plain proportional controller(solid line) and that of a PD controller (dashed line) to the samestep upset.

Controller response with and without derivation action1

0 1 2 3 4 5Time

6 7 8 9 10

0.8

−0.8

0.6

−0.6

0.4

−0.4

0.2

Cont

rolle

r out

put

−0.2

0


110 Control Theory

Only the integral term provides any input to the controlleroutput; only the integrator drives the manipulated variable tocompensate for the area under the past error curve.

In summary, the PID controller produces an outputdefined as

2.1(14)

where Kp is the proportional gain; Td is the derivative time;and Ti is the integral time.

The integral time can be viewed as the amount of thetime it takes for the integral component to make the samecontribution as the proportional term. If the integral time isshort, the integral contribution to the PID output is large andtoo much integral gain (Ti too small) can cause the systemto become unstable.

Feedforward Control

Feedforward control is another basic technique used to com-pensate for uncontrolled disturbances entering the controlledprocess. Both feedback and feedforward control are dis-cussed in detail in Section 2.8 and therefore only an intro-duction is given here. In this technique the control action isbased on the disturbance input into the process without con-sidering the condition of the process. In concept, feedforwardcontrol yields much faster correction than feedback controldoes, and in the ideal case compensation is applied in sucha manner that the effect of the disturbance is never seen inthe controlled variable, the process output.

A skillful operator of a direct contact water heater coulduse a simple feedforward strategy to compensate for changesin inlet water temperature by detecting a change in inlet watertemperature and in response to that, increasing or decreasingthe steam rate to counteract the change (Figure 2.1dd). Thissame compensation could be made automatically with aninlet temperature detector designed to initiate the appropriatecorrective adjustment in the steam valve opening.

The concept of feedforward control is very powerful, butunfortunately it is difficult to implement in most process

control applications. In many cases disturbances cannot beaccurately measured, and therefore pure feedforward cannotbe used. The main limitation of feedforward is due to ourinability to prepare perfect process models or to make per-fectly accurate measurements.

Because of these limitations, pure feedforward wouldaccumulate the errors in its model and would eventually “self-destruct.” The main limitations of feedback control are thatfeedback cannot anticipate upsets but can only respond tothem after the upsets have occurred, and that it makes itscorrection in an oscillating, cycling manner.

It has been found that combining feedback and feedfor-ward is desirable in that the imperfect feedforward modelcorrects for about 90% of the upset as it occurs, while feed-back corrects for the remaining 10%. With this approach, thefeedforward component is not pushed beyond its abilities,while the load on the feedback loop is reduced by an orderof magnitude, allowing for much tighter control.

Feedforward Response

Ideally the feedforward correction would be so effective thata disturbance would have no measurable effect on the con-trolled variable, the process output. As an example, con-sider a first-order system in which there is a measurabledisturbance. Suppose that a process disturbance occurs at timet = 5 seconds, as shown in the top segment of Figure 2.1ee, andcauses the PID controller to generate a corrective action asshown in lower part of Figure 2.1ee. Note that while thecontroller will eliminate the disturbance, it will do that onlyafter it has occurred.

FIG. 2.1ddThe concept of feedforward control implemented by a humanoperator.

u t K e tT

e d Tde t

dtpi

d( ) ( ) ( )( )= + +

∫1 τ τ

Steam

Cool waterSteamvalve

Waterheater

Temperatureindicator

Hot water

FIG. 2.1ee If a step disturbance occurs at t = 5, the controlled variable of afirst-order process responds to that upset as shown in the top portionof the figure. The bottom part shows the response of a feedbackPID controller to such an upset, which generates the manipulatedvariable.

1.5

1

0.5

00

Am

plitu

deA

mpl

itude

1 2 3 4 5

Controller output with disturbance

Process output with disturbance

6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

2

1.5

1

0.5

0



Now consider a control system that includes a 100%effective feedforward controller as shown in Figure 2.1ff. Thebottom part of Figure 2.1gg shows the response of a perfectfeedforward compensator to an upset that has occurred att = 5. The middle section of the figure shows the feedbackPID output and the top portion shows the controlled variable,which remains undisturbed.

Note that the compensator does not respond until thedisturbance occurs at time t = 5 seconds. With the feedfor-ward compensation that is added to the PID controller output

(Figure 2.1ff), the system exactly compensates for the dis-turbance. In practice, exact compensation is not possible.However, much of the effect of a disturbance can be mitigatedby a judiciously designed feedforward compensator.

Cascade Control

Section 2.6 discusses cascade control in detail; therefore,only an introduction is given here. In cascade control twocontrollers are used to control a single process variable. Forexample, consider the composition control of the productfrom a continuously stirred reactor in which an exothermicchemical reaction takes place. If the compensation is directlyrelated to the temperature within the reactor and if the com-position itself is difficult to measure online, then the reactortemperature is often chosen as the controlled variable. Tocontrol that temperature, the cooling water flow to the jacketis usually manipulated by the master (outer) temperaturecontroller, which is adjusting the set point of the slave (inner)flow controller. Together these two controllers form a cas-caded control loop, as shown in Figure 2.1hh.

The control loop for the cooling water flow is called theinner loop. The outer loop is the controller for the reactortemperature. The inner loop must have a response time thatis faster than that of the outer loop. The outer loop controllerprovides the set point for the inner loop. The inner loop mustbe faster than the outer loop, so that the set point will notchange too fast for the inner loop dynamics to follow; if itdid, stability problems would result.

Filtering

Filters are needed to eliminate noise from such signals as thecontrolled variable. If the noise in the measurement signal isnot reduced, it can pass through the controller and causecycling and eventual damage to the control valve or otherfinal control element.

Noise tends to be of a high frequency and changes morerapidly in time than does the controlled process variable. Thetask of a filter is to block the rapidly changing componentof the signal but pass the slowly changing component, so thatthe filter output will have less noise than does the raw signal.

Filtering may have little or no effect on control perfor-mance if the closed-loop system response time is slow com-pared to the response time of the filter. In other words, if the

FIG. 2.1ff Block diagram representation of a combination of a feedback PIDcontrol loop with a feedforward compensator.

FIG. 2.1ggThe response of the control loop shown in Figure 2.1ff to a processupset that occurs at t = 5. In the bottom part, the output of thefeedforward compensator is shown. When this is added to the outputof the feedback PID controller, the upset is fully corrected and asshown in the top curve, the controlled variable is not upset at all.

r e y

+ −

++

+

+

Disturbance

PID Plant

d(t)

Feedforwardcontroller

1.51

0.50

0 1 2 3 4 5

Process output with disturbance

Controller output with disturbance

Feedforward compensator output with disturbance

Time in seconds

6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

21.5

Am

plitu

deA

mpl

itude

10.5

1

0.5

0

FIG. 2.1hhBlock diagram of a cascade control loop.

PIDr

+ − −

+ePID Flow

system Reactory


112 Control Theory

filter reaction time is assumed to be instantaneous comparedto the speed of response of the closed-loop system, then thereis no dynamic effect on the closed-loop system. On the otherhand, if the closed-loop is fast relative to the speed of responseof the filter, then the dynamic effects of the filter will have aneffect.

Therefore, filtering can complicate the controller and canlimit the response of a PID controller. One way to compensatefor this is to tune the controller with the filter inserted into thefeedback loop. Although the PID controller may give betterperformance without the filter, the effectiveness of the filterdepends on the nature and amount of the noise and the desiredclosed-loop dynamics.

ADVANCED CONTROLS

The scope of the discussion in this first section was limitedto basics of process control, and therefore such subjects asmultivariable loops, interaction and decoupling, optimiza-tion, artificial intelligence, statistical process control, relativegain calculations, neural networks, model-based and model-free controls, adaptive control, fuzzy logic and statisticalprocess control have not been covered. These topics andmany others are discussed in the sections that follow.

References

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2. Shinskey, F. G., Process Control Systems, 4th ed., New York: McGraw-Hill, 1996.

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