Mikles J., Fikar M. Process Modelling, Identification, and Control
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XII Acknowledgements
Parts of the book were prepared during the stays of both authors at Ruhr
University in Bochum that were supported by the Alexander von Humboldt
Foundation. This support is very gratefully acknowledged. In this context we
particularly thank to prof. Unbehauen from the Ruhr University in Bochum
for his support, cooperation, and comments to the book.
Bratislava, J´an Mikleˇs
February 2007 Miroslav Fikar
Contents
1 Introduction ............................................... 1
1.1 TopicsinProcessControl................................. 1
1.2 An ExampleofProcessControl ........................... 3
1.2.1 Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Steady-State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.3 Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.4 Dynamical Properties of the Process . . . . . . . . . . . . . . . . . 4
1.2.5 Feedback Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.6 Transient Performance of Feedback Control . . . . . . . . . . . 6
1.2.7 Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.8 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 DevelopmentofProcessControl........................... 10
1.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Mathematical Modelling of Processes ...................... 13
2.1 General Principles of Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Examples of Dynamic Mathematical Models . . . . . . . . . . . . . . . . 16
2.2.1 Liquid Storage Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Heat Transfer Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Mass Transfer Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.4 Chemical and Biochemical Reactors . . . . . . . . . . . . . . . . . 31
2.3 General Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 Systems, Classification of Systems . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.7 Exercises ............................................... 47
3 Analysis of Process Models ................................ 51
3.1 TheLaplace Transform .................................. 51
3.1.1 Definition of the Laplace Transform . . . . . . . . . . . . . . . . . 51
3.1.2 Laplace Transforms of Common Functions . . . . . . . . . . . . 53
XIV Contents
3.1.3 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . 57
3.1.4 Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1.5 Solution of Linear Differential Equations
byLaplaceTransformTechniques ................... 62
3.2 State-Space Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 Concept of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.2 Solution of State-Space Equations . . . . . . . . . . . . . . . . . . . 67
3.2.3 Canonical Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2.4 Stability, Controllability, and Observability
of Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.5 Canonical Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3 Input-Output Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3.1 SISO Continuous Systems with Constant Coefficients . . 84
3.3.2 Transfer Functions of Systems with Time Delays . . . . . . 93
3.3.3 Algebra of Transfer Functions for SISO Systems . . . . . . . 96
3.3.4 Input Output Models of MIMO Systems - Matrix
of Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.3.5 BIBO Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.3.6 Transformation of I/O Models into State-Space Models 104
3.3.7 I/O Models of MIMO Systems - Matrix Fraction
Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.5 Exercises ...............................................114
4 Dynamical Behaviour of Processes .........................117
4.1 Time Responses of Linear Systems to Unit Impulse and Unit
Step ...................................................117
4.1.1 Unit Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1.2 Unit Step Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2.1 The Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.2.2 The Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.2.3 Runge-Kutta Method for a System of Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.2.4 Time Responses of Liquid Storage Systems . . . . . . . . . . . 135
4.2.5 Time Responses of CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.3 Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.3.1 Response of the Heat Exchanger to Sinusoidal Input
Signal............................................145
4.3.2 Definition of Frequency Responses . . . . . . . . . . . . . . . . . . . 149
4.3.3 Frequency Characteristics of a First Order System . . . . . 153
4.3.4 Frequency Characteristics of a Second Order System . . . 156
4.3.5 Frequency Characteristics of an Integrator . . . . . . . . . . . . 158
4.3.6 Frequency Characteristics of Systems in a Series . . . . . . . 159
4.4 Statistical Characteristics of Dynamic Systems . . . . . . . . . . . . . . 162
Contents XV
4.4.1 Fundamentals of Probability Theory . . . . . . . . . . . . . . . . . 162
4.4.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.4.3 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.4.4 White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.4.5 Response of a Linear System to Stochastic Input . . . . . . 178
4.4.6 Frequency Domain Analysis of a Linear System
with Stochastic Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.6 Exercises ...............................................187
5 Discrete-Time Process Models .............................189
5.1 Computer Controlled and Sampled Data Systems . . . . . . . . . . . . 189
5.2 Z – Transform ..........................................195
5.3 Discrete-Time Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.4 Input-Output Discrete-Time Models – Difference Equations . . . 204
5.4.1 Direct Digital Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.5 State-Space Discrete-Time Models . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.6 Properties of Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . 211
5.6.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5.6.2 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5.6.3 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5.6.4 Discrete-Time Feedback Systems – Control Performance 212
5.7 Examples of Discrete-Time Process Models . . . . . . . . . . . . . . . . . 214
5.7.1 Discrete-Time Tank Model . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.7.2 Discrete-Time Model of Two Tanks in Series . . . . . . . . . . 215
5.7.3 Steady-State Discrete-Time Model of Heat Exchangers
in Series..........................................216
5.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
5.9 Exercises ...............................................218
6 Process Identification ......................................221
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.1.1 Models of Linear Dynamic Systems . . . . . . . . . . . . . . . . . . 223
6.2 Identification from Step Responses . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.2.1 First Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
6.2.2 Underdamped Second Order System . . . . . . . . . . . . . . . . . 227
6.2.3 System of a Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6.3 LeastSquares Methods...................................233
6.3.1 Recursive Least Squares Method . . . . . . . . . . . . . . . . . . . . 235
6.3.2 Modifications of Recursive Least Squares . . . . . . . . . . . . . 241
6.3.3 Identification of a Continuous-time Transfer Function . . 245
6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
6.5 Exercises ...............................................251
XVI Contents
7 The Control Problem and Design of Simple Controllers . . . . 253
7.1 Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.1.1 Feedback Control Problem Definition . . . . . . . . . . . . . . . . 255
7.2 Steady-State Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.3 Control PerformanceIndices ..............................257
7.3.1 Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
7.3.2 Integral Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.3.3 Control Quality and Frequency Indices . . . . . . . . . . . . . . . 260
7.3.4 Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
7.4 PID Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
7.4.1 Description of Components . . . . . . . . . . . . . . . . . . . . . . . . . 266
7.4.2 PID Controller Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 274
7.4.3 Setpoint Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
7.4.4 Simple Rules for Controller Selection . . . . . . . . . . . . . . . . 276
7.4.5 Practical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
7.4.6 Controller Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
7.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
7.6 Exercises ...............................................294
8 Optimal Process Control ..................................297
8.1 Problem of Optimal Control and Principle of Minimum . . . . . . 297
8.2 Feedback Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
8.3 Optimal Tracking, Servo Problem, and Disturbance Rejection . 317
8.3.1 Tracking Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
8.3.2 Servo Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
8.3.3 LQ Control with Integral Action . . . . . . . . . . . . . . . . . . . . 321
8.4 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
8.4.1 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 322
8.4.2 Dynamic Programming for Discrete-Time Systems . . . . . 329
8.4.3 Optimal Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
8.5 Observers and State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
8.5.1 State Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
8.5.2 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
8.6 Analysis of State Feedback with Observer and Polynomial
PolePlacement .........................................340
8.6.1 Properties of State Feedback with Observer . . . . . . . . . . . 340
8.6.2 Input-Output Interpretation of State Feedback
with Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
8.6.3 Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
8.6.4 Polynomial Pole Placement Control Design . . . . . . . . . . . 354
8.6.5 Integrating Behaviour of Controller . . . . . . . . . . . . . . . . . . 355
8.6.6 Polynomial Pole Placement Design for Multivariable
Systems..........................................361
8.7 The Youla-Kuˇcera Parametrisation . . . . . . . . . . . . . . . . . . . . . . . . 365
8.7.1 Fractional Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 366
Contents XVII
8.7.2 Parametrisation of Stabilising Controllers . . . . . . . . . . . . 368
8.7.3 Parametrised Controller in the State-Space
Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
8.7.4 Parametrisation of Transfer Functions of the
Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
8.7.5 Dual Parametrisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
8.7.6 Parametrisation of Stabilising Controllers for
Multivariable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
8.7.7 Parametrisation of Stabilising Controllers for
Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
8.8 Observer LQ Control, State-Space and Polynomial
Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
8.8.1 Polynomial LQ Control Design with Observer
forSinglevariableSystems ..........................381
8.8.2 Polynomial LQ Design with State Estimation
for Multivariable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 385
8.9 LQG Control, State-Space and Polynomial Interpretation . . . . . 388
8.9.1 Singlevariable Polynomial LQG Control Design . . . . . . . 388
8.9.2 Multivariable Polynomial LQG Control Design . . . . . . . . 392
8.10 H
2
Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
8.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
8.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
9 Predictive Control .........................................403
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
9.2 IngredientsofMBPC ....................................404
9.2.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
9.2.2 Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
9.3 Derivation and Implementation of Predictive Control . . . . . . . . 407
9.3.1 Derivation of the Predictor . . . . . . . . . . . . . . . . . . . . . . . . . 407
9.3.2 Calculation of the Optimal Control . . . . . . . . . . . . . . . . . . 409
9.3.3 Closed-Loop Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
9.3.4 Derivation of the Predictor from State-Space Models . . . 412
9.3.5 Multivariable Input-Output Case . . . . . . . . . . . . . . . . . . . . 416
9.3.6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
9.3.7 Relation to Other Approaches . . . . . . . . . . . . . . . . . . . . . . 418
9.3.8 Continuous-Time Approaches . . . . . . . . . . . . . . . . . . . . . . . 418
9.4 ConstrainedControl .....................................419
9.5 Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
9.5.1 Stability Results in GPC . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
9.5.2 Terminal Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
9.5.3 Infinite Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
9.5.4 Finite Terminal Penalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
9.6 Explicit PredictiveControl ...............................426
9.6.1 Quadratic Programming Definition . . . . . . . . . . . . . . . . . . 426
XVIII Contents
9.6.2 Explicit Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
9.6.3 Multi-Parametric Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . 429
9.7 Tuning.................................................431
9.7.1 Tuning based on the First Order Model . . . . . . . . . . . . . . 431
9.7.2 Multivariable Tuning based on the First Order Model . . 432
9.7.3 Output Horizon Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
9.7.4 λ Tuning.........................................434
9.7.5 Tuning based on Model Following . . . . . . . . . . . . . . . . . . . 434
9.7.6 The C polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
9.8 Examples...............................................436
9.8.1 A Linear Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
9.8.2 Neural Network based GPC . . . . . . . . . . . . . . . . . . . . . . . . 438
9.8.3 pH Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
9.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
9.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
10 Adaptive Control ..........................................445
10.1 Discrete-Time Self-Tuning Control . . . . . . . . . . . . . . . . . . . . . . . . . 446
10.2 Continuous-Time Self-Tuning Control . . . . . . . . . . . . . . . . . . . . . . 447
10.3 Examples of Self-Tuning Control . . . . . . . . . . . . . . . . . . . . . . . . . . 447
10.3.1 Discrete-Time Adaptive Dead-Beat Control of a
Second Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
10.3.2 Continuous-Time Adaptive LQ Control of a Second
OrderSystem.....................................451
10.3.3 Continuous-Time Adaptive MIMO Pole Placement
Control ..........................................456
10.3.4 Adaptive Control of a Tubular Reactor . . . . . . . . . . . . . . . 459
10.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
References .....................................................465
Index ..........................................................475
List of Figures
1.1 Asimple heat exchanger ................................. 3
1.2 Response of the process controlled with proportional
feedback controller for a step change of disturbance variable ϑ
v
7
1.3 The scheme of the feedback control for the heat exchanger . . . 8
1.4 The block scheme of the feedback control of the heat exchanger 9
2.1 Aliquid storagesystem .................................. 16
2.2 Aninteractingtank-in-seriesprocess....................... 17
2.3 Continuous stirred tank heated by steam in jacket . . . . . . . . . . . 19
2.4 Seriesofheatexchangers................................. 21
2.5 Double-pipe steam-heated exchanger and temperature profile
along the exchanger length in steady-state . . . . . . . . . . . . . . . . . . 23
2.6 Temperature profile of ϑ in an exchanger element of length
dσ for time dt .......................................... 23
2.7 Ametalrod ............................................ 24
2.8 A scheme of a packed countercurrent absorption column . . . . . 26
2.9 Scheme of a continuous distillation column . . . . . . . . . . . . . . . . . 28
2.10 Model representation of i-thtray.......................... 29
2.11 A nonisothermal CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.12 Classification of dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . 46
2.13 A cone liquid storage process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.14 Well mixed heat exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.15 A well mixed tank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.16 Series of two CSTRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.17 A gas storage tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1 A step function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 An original and delayed function . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3 A rectangular pulse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Amixing process........................................ 68
3.5 AU-tube .............................................. 74
XX List of Figures
3.6 Time response of the U-tube for initial conditions (1, 0)
T
..... 76
3.7 Constant energy curves and state trajectory of the U-tube in
the state plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.8 Canonical decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.9 Block scheme of a system with transfer function G(s)........ 85
3.10 Two tanks in a series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.11 Block scheme of two tanks in a series . . . . . . . . . . . . . . . . . . . . . . 88
3.12 Serial connection of n tanks .............................. 90
3.13 Block scheme of n tanks in aseries ........................ 90
3.14 Simplified block scheme of n tanksin aseries............... 91
3.15 Block scheme of a heat exchanger . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.16 Modified block scheme of a heat exchanger . . . . . . . . . . . . . . . . . 92
3.17 Block scheme of a double-pipe heat exchanger . . . . . . . . . . . . . . 97
3.18 Serial connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.19 Parallel connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.20 Feedback connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.21 Moving of the branching point against the direction of signals 99
3.22 Moving of the branching point in the direction of signals . . . . . 99
3.23 Moving of the summation point in the direction of signals . . . . 100
3.24 Moving of the summation point against the direction of signals 100
3.25 Block scheme of controllable canonical form of a system . . . . . . 106
3.26 Block scheme of controllable canonical form of a second
ordersystem ...........................................107
3.27 Block scheme of observable canonical form of a system . . . . . . . 108
4.1 Impulse response of thefirstordersystem ..................119
4.2 Stepresponseof afirstorder system.......................120
4.3 Step responses of a first order system with time constants
T
1
,T
2
,T
3
...............................................121
4.4 Step responses of the second order system for the various
values of ζ .............................................123
4.5 Step responses of the system with n equal time constants . . . . 124
4.6 Block scheme of the n-th order system connected in a series
with time delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.7 Step response of the first order system with time delay . . . . . . . 125
4.8 Step response of the second order system with the numerator
B(s)=b
1
s +1..........................................125
4.9 Simulinkblockscheme...................................133
4.10 Results from simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.11 Simulink block scheme for the liquid storage system . . . . . . . . . 137
4.12 Response of the tank to step change of q
0
..................138
4.13 Simulink block scheme for the nonlinear CSTR model . . . . . . . 143
4.14 Responses of dimensionless deviation output concentration
x
1
to step change of q
c
...................................146
List of Figures XXI
4.15 Responses of dimensionless deviation output temperature x
2
to step change of q
c
.....................................146
4.16 Responses of dimensionless deviation cooling temperature x
3
to step change of q
c
.....................................147
4.17 Ultimate response of the heat exchanger to sinusoidal input . . 149
4.18 The Nyquist diagram for the heat exchanger . . . . . . . . . . . . . . . . 153
4.19 The Bode diagram for the heat exchanger . . . . . . . . . . . . . . . . . . 154
4.20 Asymptotes of the magnitude plot for a first order system . . . . 155
4.21 Asymptotes of phase angle plot for a first order system . . . . . . 156
4.22 Asymptotes of magnitude plot for a second order system . . . . . 157
4.23 Bode diagrams of an underdamped second order system
(Z
1
=1,T
k
=1) ........................................158
4.24 The Nyquist diagram of an integrator . . . . . . . . . . . . . . . . . . . . . . 159
4.25 Bode diagram of an integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.26 The Nyquist diagram for the third order system . . . . . . . . . . . . . 161
4.27 Bode diagram for the third order system . . . . . . . . . . . . . . . . . . . 161
4.28 Graphical representation of the law of distribution of a
random variable and of the associated distribution function . . . 163
4.29 Distribution function and corresponding probability density
function of a continuous random variable . . . . . . . . . . . . . . . . . . . 166
4.30 Realisations of a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . 170
4.31 Power spectral density and auto-correlation function of white
noise ..................................................177
4.32 Power spectral density and auto-correlation function of the
process given by (4.276) and (4.277) . . . . . . . . . . . . . . . . . . . . . . . 178
4.33 Block-scheme of a system with transfer function G(s)........183
5.1 A digital control system controlling a continuous-time process 189
5.2 Transformation of a continuous-time to a discrete-time signal . 190
5.3 Continuous-time signal step reconstruction from its
discrete-time counterpart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.4 A possible loss of information caused by sampling . . . . . . . . . . . 192
5.5 Anideal sampler........................................192
5.6 Spectral density of a signal y(t) ..........................193
5.7 Spectral density of a sampled signal y
∗
(t) ..................194
5.8 An ideal sampler and an impulse modulated signal . . . . . . . . . . 195
5.9 Sample-and-hold device in series with a continuous-time system201
5.10 Sampler and zero-order hold in series with a continuous-time
system G(s) ...........................................202
5.11 Direct digital control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.12 Zero-order hold in series with a continuous-time system . . . . . . 207
5.13 A feedback discrete-time system . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5.14 Series of heat exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
6.1 Stepresponse of thefirstorder system .....................226