Predictive Control in Process Engineering - Robert Haber - E-Book

Predictive Control in Process Engineering E-Book

Robert Haber

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Beschreibung

Describing the principles and applications of single input, single output and multivariable predictive control in a simple and lively manner, this practical book discusses topics such as the handling of on-off control, nonlinearities and numerical problems. It gives guidelines and methods for reducing the computational demand for real-time applications. With its many examples and several case studies (incl. injection molding machine and waste water treatment) and industrial applications (stripping column, distillation column, furnace) this is invaluable reading for students and engineers who would wish to understand and apply predictive control in a wide variety of process engineering application areas.

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Contents

Cover

Half Title page

Title page

Copyright page

Preface

Acknowledgements

References

Notation and Abbreviations

Chapter 1: Introduction to Predictive Control

1.1 Preview of Predictive Control

1.2 Manipulated, Reference, and Controlled Signals

1.3 Cost Function of Predictive Control

1.4 Reference Signal and Disturbance Preview, Receding Horizon, One-Step-Ahead, and Long-Range Optimal Control

1.5 Free and Forced Responses of the Predicted Controlled Variable

1.6 Minimization of the Cost Function

1.7 Simple Tuning Rules of Predictive Control

1.8 Control of Different Linear SISO Processes

1.9 Control of Different Linear MIMO Processes

1.10 Control of Nonlinear Processes

1.11 Control under Constraints

1.12 Robustness

1.13 Summary

References

Chapter 2: Linear SISO Model Descriptions

2.1 Nonparametric System Description

2.2 Pulse-Transfer Function Model

2.3 Discrete-Time State Space Model

2.4 Summary

References

Chapter 3: Predictive Equations of Linear SISO Models

3.1 Predictive Equations Based on Nonparametric Models

3.2 Predictive Equations Based on the Pulse-Transfer Function

3.3 Predictive Equations of the State Space Model

3.4 Summary

References

Chapter 4: Predictive On–Off Control

4.1 Classical On–Off Control by Means of Relay Characteristics

4.2 Predictive Set Point Control

4.3 Predictive Start-Up Control at a Reference Signal Change

4.4 Predictive Gap Control

4.5 Case Study: Temperature Control of an Electrical Heat Exchanger

4.6 Summary

References

Chapter 5: Generalized Predictive Control of Linear SISO Processes

5.1 Control Algorithm without Constraints

5.2 Linear Polynomial Form of Unconstrained GPC

5.3 Tuning the Controller Parameters

5.4 Blocking and Coincidence Points Techniques

5.5 Measured Disturbance Feed-Forward Compensation

5.6 Control Algorithm with Constraints

5.7 Extended GPC with Terminal Methods

5.8 Summary

References

Chapter 6: Predictive PID Control Algorithms

6.1 Predictive PI(D) Control Structure

6.2 Predictive PI Control Algorithm

6.3 Predictive PID Control Algorithm

6.4 Equivalence between the Predictive PI(D) Algorithm and the Generalized Predictive Control Algorithm

6.5 Tuning of Predictive PI(D) Algorithms

6.6 Robustifying Effects Applied for Predictive PI(D) Control Algorithms

6.7 Summary

References

Chapter 7: Predictive Control of Multivariable Processes

7.1 Model Descriptions

7.2 Predictive Equations

7.3 The Control Algorithm

7.4 Polynomial Form of the Controller (without Matrix Inversion)

7.5 Pairing of the Controlled and the Manipulated Variables

7.6 Scaling of the Controlled and the Manipulated Variables

7.7 Tuning

7.8 Decoupling Control

7.9 Case Study: Control of a Distillation Column

7.10 Summary

References

Chapter 8: Estimation of the Predictive Equations

8.1 LS Parameter Estimation

8.2 More-Steps-Ahead Prediction Based on the Estimated Process Model

8.3 Long-Range Optimal Single Process Model Identification

8.4 Multi-Step-Ahead Predictive Equation Identification

8.5 Comparison of the Long-Range Optimal Identification Algorithms

8.6 Case Study: Level Control in a Two-Tank Plant

8.7 Summary

References

Chapter 9: Multimodel and Multicontroller Approaches

9.1 Nonlinear Process Models

9.2 Predictive Equations

9.3 The Control Algorithm

9.4 Case Study

9.5 Summary

References

Chapter 10: GPC of Nonlinear SISO Processes

10.1 Nonlinear Process Models

10.2 Predictive Equations for the Nonparametric and Parametric Hammerstein and Volterra Models

10.3 Control Based on Nonparametric and Parametric Hammerstein and Volterra Models

10.4 Control Based on Linearized Models

10.5 Control Based on Nonlinear Free and Linearized Forced Responses

10.6 Case Study: Level Control of a Two-Tank Plant

10.7 Summary

References

Chapter 11: Predictive Functional Control

11.1 Control Strategy and Controller Parameters for a Constant Set Point

11.2 PFC for Aperiodic Processes

11.3 PFC with Disturbance Feed-Forward

11.4 PFC with Constraints

11.5 Nonlinear PFC for Processes with Signal-Dependent Parameters

11.6 Case Study: Temperature Control of a Hot Air Blower

11.7 Summary

References

Chapter 12: Case Studies

12.1 Predictive Temperature Control of an Injection Molding Machine

12.2 Wastewater Quality Control of an Intermittently Operated Plant

12.3 Wastewater Quality Control with Pre-denitrification

References

Chapter 13: Industrial Applications

13.1 Concentration Control

13.2 Concentration Control and Reducing Steam Consumption

13.3 Temperature and Combustion Control

References

Chapter 14: Practical Aspects and Future Trends

14.1 Classification of a Predictive Control Project

14.2 Project Implementation

14.3 Implementation of a Predictive Controller

14.4 Future Trends

14.5 Summary

References

Index

Robert Haber, Ruth Bars, and Ulrich Schmitz

Predictive Control in Process Engineering

Related Titles

Abel, D., Epple, U., Spohr, G.-U. (eds.)Integration von Advanced Control in der ProzessindustrieRapid Control Prototyping 2008 ISBN: 978-3-527-31205-4

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The Authors

Prof. Robert HaberCologne University of Applied SciencesInstitute of Plant & Process EngineeringBetzdorfer Str. 250679 KlnGermanyrobert.haber@fh-koeln.de

Prof. Ruth BarsBudapest University of Technology &EconomicsDepartment of Automationand Applied InformaticsMagyar Tudsok Krtja 21117 BudapestHungarybars@aut.bme.hu

Ulrich SchmitzShell Deutschland Oil GmbHRheinland Raffinerie GodorfLudwigshafener Strae 150389 [email protected]

Cover illustrationShell Rheinland Refinery, with permission

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Preface

The predictive control idea emerged in the 1970s as an industrial approach to process control. Today this technique is the most frequently applied advanced process control (APC) method in industry.

Basic control algorithms are often extended by advanced control algorithms to exploit a plant better, to increase the benefits, to reduce the costs and manpower, and so on. To the advanced algorithms belong

advanced regulatory control (e.g., cascade, ratio, override, disturbance feed-forward control, fuzzy logic control, etc.)model-based control (e.g., predictive control, internal model control, Smith predictor, etc.)real-time optimization (linear or nonlinear, static or dynamic optimization).

Only the last two methods are considered as APC methods. Common to these techniques is that they are model-based algorithms, that is, a model of the process (and often of the disturbances) is required and used during online calculations.

Figure 1 shows the hierarchy of the four control levels. It is seen that the achievable benefits are greatest with basic control. Therefore, it is very important that the sensors and the valves be of good quality and the basic controllers be tuned well. On the other hand, APC applications help to improve the efficiency, which could be achieved otherwise only by extending and modifying the plant. (A similar but more detailed hierarchy is shown in Blevins et al. [1].)

Figure 1 Necessary effort to build a hierarchical control structure and the achievable benefits.

The capital costs as a function of the achievable benefits are shown in Figure 2.

Figure 2 Benefits of control and optimization.

Source: ARC Advisory Group, Real-time Process Optimization & Training Outlook, 2008.

APC applications have mostly been applied in the refining and petrochemical industry; see, for example, the statistics in Figure 3 presented by Qin and Badgwell [2]. This diagram is based only on the implementations of two big vendors, Aspen Technology and Honeywell.

Figure 3 APC applications by area of operation.

On the basis of process models which can be obtained by simple measurements even in an industrial environment, predictive control provides good tracking and disturbance rejection behavior considering also constraints for both single-input, single-output (SISO) and multi-input, multi-output (MIMO) systems. In recent years, besides the generally used proportional plus integral plus derivative (PID) control algorithms, predictive control has gained significant industrial acceptance, supported also by industrial software. In the case of a long dead time and a known reference trajectory, predictive control ensures much faster performance than the PID algorithms.

Figure 4 shows statistics based on about 150 responses to a questionnaire from the ARC Advisory Group (founded as Automation Research Corporation) related to the industry sectors refinery, chemicals, oil and gas, power, pulp and paper, polymers, metals, mining, food and beverage, cement, glass, and pharmaceuticals.

Figure 4 Usage of model-based predictive control.

Source: ARC Advisory Group, Real-time Process Optimization & Training Outlook, 2008.

Bauer and Craig [3] surveyed the answers of 38 APC users and 28 APC suppliers to a questionnaire about APC applications. Figure 5 shows the statistics of the “standard” and “frequently” used advanced techniques. (The possible categories for how frequently the method was used were in decreasing order: standard, frequently, rarely, and never. As the answers to the questionnaire may be subjective, the statistics presented should be interpreted with caution.) Though not all techniques listed belong to APC (e.g., split-range control), and some applications might have used several of the methods listed (e.g., model-based predictive control (MPC) based on a neural net model, or constrained control by means of MPC), the diagram shows that today MPC is the most frequently applied APC method in industry. Further, the survey of Bauer and Craig confirms the finding of the survey of Qin and Badgwell, namely, the refinery, petrochemical, and chemical industries are the largest users of APC. As a fourth sector, mineral processing was mentioned by Bauer and Craig.

Figure 5 Most frequently used advanced control methods in industry.

Figure 6 shows the level of adoption of APC by industrial users based on the same questionnaire evaluated in Figure 2. The diagram shows that APC is rather applied to large plants than to small or medium-sized plants (because of the higher achievable profits and safety requirements with larger plants). The acceptance is best with the leaders (companies with high technology), not as good with competitors, and even lower with followers. The last bar “all” shows the average value.

Figure 6 Current level of APC implementations.

The aim of this book is to introduce the topic of predictive control, to give a detailed discussion of the control algorithms for both linear SISO and linear MIMO systems, to describe some predictive control methods for nonlinear systems, and also to discuss some new trends and practical aspects in predictive control. The book intends to make the topic understandable and practically applicable for students and industrial users by giving the philosophy of the control ideas and also the mathematics behind them with computational details. The book was written for senior undergraduate and graduate students and for engineers who would like to apply advanced control techniques in industrial practice. The book supposes basic knowledge of control theory.

A simple introduction and a discussion of predictive control concepts with straightforward explanations are given. The theoretical results are demonstrated through simple examples. The details of the calculations are described, and simulations illustrate the behavior of the algorithms. The reader will get some practice in applying predictive control and an insight into the effects of the tuning parameters.

The control algorithms are based on adequate system models. These models can be obtained by first principles describing the system behavior or by identification based on input/output (I/O) measurements. Here the systems are represented by discrete-time, mainly by I/O models. In some cases state space models are also considered. As predictive control requires predictive process models, providing prediction of future outputs based on information available till the current time point, generally predictive transformation of the model equations is required. Some aspects of identification of the parameters of the predictive models themselves, that is, some details of identification for predictive control, are also dealt with.

In control practice on–off control is widely applied providing simple and cheap control solutions. On the other hand, predictive algorithms are very rarely presented for on–off valves. Different predictive solutions for on–off control have been developed and are discussed in this book.

Nowadays, nonlinear predictive control solutions are at the forefront of interest. As generally the systems are nonlinear, control algorithms considering the nonlinearities would provide better performance in the whole operating range than linear algorithms. Some new predictive algorithms for control of nonlinear processes have been worked out and their performance has been analyzed. Here, predictive control algorithms based on the Hammerstein and Volterra model approximation of the system model and solutions using multimodel and multicontroller approaches are discussed and their performance is demonstrated.

For real-time applications the computational demand of the control algorithms is of significant importance. Several possibilities to reduce the computation time are dealt with. The algorithms applying optimization procedures under constraints could be applied rather in the case of slow processes.

The main criticism of predictive control was that in its original formulations stability is not guaranteed. Stability and robustness issues have been addressed in the technical literature recently and robust predictive control solutions can be obtained. Some predictive control solutions guaranteeing stability are also given, but for detailed discussion on stability and robustness, we refer to the literature.

Experiences with an industrial predictive control program package, several simulated case studies (e.g., injection molding machine, wastewater treatment), and some applications from a refinery (distillation column, gas-heated furnace) are given. Several practical aspects based on our experiences and based on the literature are presented.

The book consists of 14 chapters. Chapter 1 gives an introduction to predictive control. The control algorithms are based on adequate system models. Chapter 2 presents the different linear SISO model descriptions, whereas Chapter 3 derives their predictive equations. Chapter 4 presents new conceptions and algorithms of predictive on–off control. Linear predictive control is discussed in Chapter 5. Chapter 6 shows predictive PID algorithms which equip PID control with predictive properties. Chapter 7 discusses predictive algorithms applied to MIMO systems. Chapter 8 shows identification methods to determine the coefficients of the predictive equations. Chapter 9 discusses predictive algorithms for nonlinear systems based on Hammerstein and Volterra models, whereas Chapter 10 describes multimodel and multicontroller predictive control solutions for nonlinear systems. Chapter 11 presents predictive functional control, a simply realizable very effective method. Chapter 12 gives some case studies (temperature control of an injection-molding machine, wastewater quality control of two plants, the first with an intermittently operated plant and the second with pre-denitrification). Chapter 13 discusses solutions and experiences with some industrial predictive control applications in a refinery (distillation column, gas-heated furnace). Chapter 14 discusses the methodology and practical aspects of the introduction of predictive control in an industrial environment.

A flowchart describing the organization of the book is given in Figure 7.

Figure 7 Flowchart describing the organization of the book.

In the main streamline, the usual predictive control topics are introduced and discussed. The aim of the authors was to give clear explanations and demonstrations of the methods through simple examples with computational details. For nonlinear control, simplified effective control algorithms are given as derived by the authors. In the parallel streamline, additional topics are considered, related to the main streamline and including also contributions of the authors. Estimation of predictive equations involves a procedure for how to estimate the coefficients of the predictive equations immediately instead of identifying the original parameters and then executing predictive transformation. Predictive PID control equips the well-known PID algorithms with predictive properties, introducing PID control algorithms with embedded predictive features. Nonlinear predictive control algorithms nowadays are at the forefront of interest, as considering the nonlinear characteristics of the process in the control algorithm promises better control performance in the whole operating range than using linear control algorithms. On–off control is discussed here in a predictive context. Also, nonlinear multimodel and multicontroller approaches are shown to control nonlinear processes. Case studies and some industrial applications demonstrate the effectiveness of predictive control applications. (Case studies are presented not only in Chapter 12 but also at the end of Chapters 4, 9, and 11, as shown in Figure 7.) Finally, some practical aspects of advanced control algorithms in practice and a discussion of future trends are given.

The material in this book is based on the literature on predictive control (textbooks and papers) and also on the research work of the authors. The experiences of our teaching courses have shown that explaining general ideas and research results has to be extended by background materials, explanations, and examples. It took several years to write the text, which we think can be taught at universities. The third author, Ulrich Schmitz, was a PhD student of the first two authors and contributed a lot to the practical realization and simulation of the algorithms. (He defended his thesis on the topic of nonlinear predictive control in 2006.)

The first draft of the material provided the basis for short intensive PhD courses held in the Department of Process Control, Helsinki University of Technology, Finland, in 1999, 2003, and 2006 (by invitation from Prof. Sirkka-Liisa Jämsä-Jounela) and master courses offered in the Faculty of Electrical Engineering and Informatics of Budapest University of Technology and Economics, Hungary (department of the second author), and the Faculty of Process Engineering, Energy, and Mechanical Systems of Cologne University of Applied Sciences, Germany (department of the first author).

Some lectures on selected topics in predictive control were held in the frame of master and PhD courses in the Faculty of Informatics and Electrical Engineering, University of Rostock, Germany, in 2000 (by invitation from Prof. B. Lampe), in the Faculty of Electrical Engineering and Information Technology of the Slovak University of Technology, Bratislava, Slovak Republic, in 2004 (by invitation from Prof. S. Kožak), in the Faculty of Electrical Engineering of Louisiana State University, Lafayette, USA, in 2002 (by invitation from Prof. Fahmida Chowdhury), and in the Department of Systems Engineering and Automation of the University of Seville, Spain, in 2008 (by invitation from Prof. E.F. Camacho).

Acknowledgements

This book is the result of collaboration between Cologne University of Applied Sciences, Germany, and Budapest University of Technology and Economics, Hungary, supported by the EU Socrates-Erasmus project and by the “Internationalization” fund of Cologne University of Applied Sciences. Some parts of the material included in this book are the results of research work sponsored by a grant from the Hungarian Academy of Sciences for control research and by the Hungarian National Research Fund for Control Research through grants T042741 and T068370. This work is also connected with the scientific program of the project “Development of quality-oriented and cooperative R + D + I strategy and functional model at Budapest University of Technology and Economics.” This project is supported by the New Hungary Development Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002). The development of the predictive control strategy for wastewater clarification was supported by the Federal Ministry for Education and Research (BMBF) as promotion of applied research and development, recommended by the German Federation of Industrial Cooperative Research Associations “Otto von Guericke”.

In February and March 2008, the first two authors performed research work in the Department of Systems Engineering and Automation of the University of Seville, Spain, in the frame of the Marie Curie program MOBILITY-1.3 “Improving the tuning methodology for MPC” project. The authors express their thanks for the invitation and the fruitful discussions with Prof. E.F. Camacho, C. Bordons and Dr. J. Gruber. The common research in Seville contributed to some new results on nonlinear predictive control.

We are also very grateful for the invitation by our colleagues (listed above) which allowed us to lecture on parts of the book and get feedback on the material. We highly appreciate the permanent support of our colleagues in our departments.

The Laboratory of Process Automation of Cologne University of Applied Sciences has had working contact with the Department of Technology of Shell’s Rhineland Refinery for many years. The authors gratefully acknowledge the collaborations in several diploma theses and long discussions with the engineers U. Volk and H. Golisch about predictive control of industrial processes. The third author is now working with Shell on APC applications. We tried to incorporate all these experiences in the present book.

Dr. D. Honc, from the Department of Control Systems, Institute of Electrical Engineering and Informatics, University of Pardubice, Czech Republic, has spent several months at Cologne University of Applied Sciences. He provided several valuable comments on the manuscript and the discussions with him were very helpful. He contributed to the topics of predictive control of multivariable and nonlinear processes.

Dr. M. Kvasnica from the Department of Information Engineering and Process Control of the Slovak University of Technology, Bratislava, coauthor of the Multi-Parametric Toolbox, explained to us multiparametric programming for solving constrained predictive control and helped us write the corresponding section of the book.

The authors are also grateful to Dr. J. Richalet, one of the pioneers of predictive control and a former manager at ADERSA (France), who drew our attention to the easily implementable predictive functional control. Two of the authors participated in and enjoyed his impressive training course.

The book was typeset in by B. Moddemann, a student of ours in Cologne. He did an excellent job.

All three authors thank their families for their support, patience, and understanding.

The authors hope that the material will be useful in understanding and teaching topics in advanced control and also in providing some ideas for further research. The authors would be thankful for any comments, corrections, and recommendations by the readers.

June 2011

CologneRobert Haber(robert.haber@fh-koeln.de)BudapestRuth Bars(bars@aut.bme.hu)Frechen (near Cologne)Ulrich Schmitz([email protected])

References

1 Blevins, T.L., McMillan, G.K., Wojsznis, W.K., and Brown, M.W. (2003) Advanced Control Unleashed: Plant Performance Management for Optimum Benefit, Research Triangle Park, USA: ISA (Instrumentation Systems and Automation Society).

2 Qin, S.J. and Badgwell, T.A. (2003) A survey of industrial model predictive control technology. Control Engineering Practice, 11, 733–764.

3 Bauer, M. and Craig, I.K. (2008) Economic assessment of advanced process control – a survey and framework. Journal of Process Control, 18, 2–18.

Notation and Abbreviations

Notation

A(q−1)polynomial matrix of the output signal in the matrix fraction modelAxsystem matrix of the CARMA state space descriptionAΔxsystem matrix of the CARIMA state space descriptionA(q−1)denominator polynomial of the pulse-transfer functiontwo-dimensional polynomial of quadratic output termsAΔ(q−1)denominator polynomial of (1−q−1)A(q−1)aicoefficient of the polynomial A(q−1)aΔicoefficient of the polynomial AΔ(q−1)B(q−1)polynomial matrix of input terms in the matrix fraction modelbx,Bxinput vector and matrix of the CARMA state space description (SISO, MIMO)input vector and matrix of the CARIMA state space description (SISO, MIMO)B(q−1)numerator polynomial of the pulse-transfer functiontwo-dimensional polynomial of quadratic input termsbicoefficient of the polynomial B(q−1)cx,Cxoutput vector and matrix of the CARMA state space description (SISO, MIMO)cΔx,CΔxoutput vector and matrix of the CARIMA state space description (SISO, MIMO)c0,c1,c2coefficients of quadratic polynomialsdiagdiagonal matrixddead time relative to the sampling timedijdead time relative to the sampling time from input j to output i of a MIMO processEj(q−1),Fj(q−1)polynomial matrices obtained by solving the MIMO Diophantine equationEj(q−1),Fj(q−1)polynomials obtained by solving the SISO Diophantine equationj-steps-ahead predicted signal of e(k)e(k)discrete-time control errorFcoefficient matrix of yp for prediction of polynomial of y(k) with j-steps-ahead output predictionGcoefficient matrix of Δuf for prediction of G(q−1)MIMO pulse-transfer function matrixpostcompensator pulse-transfer function matrixprecompensator pulse-transfer function matrixpulse-transfer function matrix of a process with a postcompensatorpulse-transfer function matrix of a process with a precompensator and a postcompensatorpulse-transfer function matrix of a process with a precompensatorG(jω)frequency functionG(q−1)pulse-transfer functionG(s)transfer functiongkcoefficient of the weighting function seriespulse-transfer function between input j and output i of a MIMO processGij(s)transfer function between input j and output i of a MIMO processcoefficient matrix of Δuf for prediction of Hpcoefficient matrix of Δup for prediction of coefficient matrix of for prediction of coefficient matrix of Δvm,p for prediction of matrix polynomial of current and future Δu(k) terms with j-steps-ahead (MIMO) output predictionmatrix polynomial of past Δu(k) terms with j-steps-ahead MIMO output predictionmatrix polynomial of Δu(k) with j-steps-ahead MIMO output predictionpolynomial of current and future Δu(k) terms with j-steps-ahead SISO output predictionpolynomial of past Δu(k) terms with j-steps-ahead SISO output predictionpolynomial of Δu(k) with j-steps-ahead SISO output predictioncoefficient of the step response seriesquadratic Volterra kernelIunity matrixIM×Mdiagonal unity matrix of dimension M×MJcost functionderivative of cost function J according to Δustatic gain matrix of postcompensatorstatic gain matrix of precompensatorcontroller gain matrix of the GPC algorithm in the MIMO casecontroller gain vector of the GPC algorithm in the SISO casekdiscrete timeKccontroller gainKpstatic gainKrset point weighting factorKpijstatic gain from input j to output i of a MIMO processMnumber of input and output signals of a MIMO systemmmemory of a nonparametric modelMunumber of input signals of a MIMO systemMynumber of output signals of a MIMO systemmaxmaximumminminimumnorder of a pulse-transfer functionnadegree of polynomial A(q−1)nbdegree of polynomial B(q−1)ntdegree of polynomial T(q−1)ne(extended) prediction horizon beyond the dead timene1first point of the extended prediction horizon beyond the dead timene1ifirst point of the extended prediction horizon of output i for a MIMO processne2last point of the extended prediction horizon beyond the dead timene2ilast point of the extended prediction horizon of output i for a MIMO processne,horlength of the long-range prediction horizonnijmodel order of a submodel between input j and output i of a MIMO processnulength of the control horizonPr(q−1)reference signal filterPy(q−1)controlled signal filterPijprocess between input j and output i of a MIMO processp0,p1,p2coefficients of a discrete-time PID controller1−q−1difference operatorq−1backward shift operatortwo-dimensional polynomial of product terms of input and output signalsR(q−1)polynomial of Δu(k) in the RST algorithmsargument of the Laplace transformationS(q−1)polynomial of y(k) in the RST algorithmsgn(.)two-value signum function (0 or 1)sign(.)three-value signum function (−1, 0, or 1)ΔTsampling timeT(q−1)MIMO robustness/disturbance filter matrix (filter of unmeasured noise)tcontinuous timeT(q−1)robustness/disturbance filter polynomial (filter of unmeasured noise)T1,T2time constantsTDderivative time constant (of a PID controller)Tdcontinuous-time dead timefilter time constant of the derivative part of a PID controllerTIintegrating time constant (of a PI(D) controller)ticoefficient of the polynomial T(q−1)TLlatent dead timeTr(q−1)polynomial of yr(k) in the RST algorithmTTlatent time constantTdijcontinuous-time dead time from input j to output i of a MIMO processTijtime constant from input j to output i of a MIMO processtinflinflexion time pointtsimsimulation timeu(k)discrete-time input signal (manipulated variable)Δu(k)increment of input signal (manipulated variable)Usteady-state input signalu(t)continuous-time input signal (manipulated variable)uF(k)filtered discrete-time input signalΔuF(k)filtered increment of the input signal (manipulated variable)ulow,uuplower and upper constraints of the output signalΔulow,Δuuplower and upper constraints of control signal incrementsuoff,uonmanipulated variable values with on–off controlu(k)multivariable control input u1(k),…,uM(k)Δu(k)multivariable control input increments Δu1(k),…,ΔuM(k)ufvector of current and future manipulated variablesΔufvector of current and future input signal incrementsΔupvector of past input signal incrementsΔuact(k)current manipulated variable vector (with a MIMO process)v(k)discrete-time auxiliary signalv(t)continuous-time auxiliary signalvm(k)measurable disturbanceΔvm,fvector of current and future measured disturbancesΔvm,pvector of past measured disturbancesvu(k)unmeasured disturbancej-steps-ahead predicted signal x(k)x(k)discrete-time state variable of the CARMA modelxΔ(k)discrete-time state variable of the CARIMA modelΔyvector of output signal incrementsforced response vector of output signal incrementsΔyfreefree response vector of output signal incrementslower and upper constraints of output signal incrementsj-steps-ahead prediction of signal y(k) in the SISO casej-steps-ahead predicted forced response of y(k)j-steps-ahead predicted free response of y(k)vector of predicted outputs in the prediction domainj-steps-ahead prediction of signal y(k) in the MIMO casevector of predicted forced responses in the prediction domainvector of predicted free responses in the prediction domainy(k)vector of multivariable outputs y1(k),…,yM(k)ypvector of past outputsYsteady-state output signaly(t)continuous-time output signal (controlled variable)yF(k)filtered discrete-time output signalyr(k)discrete-time reference signaldiscrete-time reference trajectorylower and upper constraints of the output signallower and upper reference value limitsy(k)discrete-time output signal (controlled variable)Δy(k)output signal increment[yr]year (dimension)[d]day (dimension)[h]hour (dimension)[s]second (dimension)Φi(.)weighting factor of the ith locally valid modelΛuweighting factor matrix of the control increments (manipulated signal changes)Λyweighting factor matrix of the control errorsλrcontrol error reduction factor (with PFC)λuweighting factor of the control increments (manipulated signal changes)scaling factor of the control increments (manipulated signal changes)λyweighting factor of the control errorscaling factor of the control errorrelative gain of controlled variable i with respect to manipulated variable jωfrequencyω0natural frequency of a second-order oscillating processσistandard deviation of the ith Gaussian validity functionξdamping factor of a second-order oscillating processθmodel parameter vectorΦmemory vector

Abbreviations

APCadvanced process controlARCadvanced regulatory controlARMAXautoregressive moving average model with exogenous inputARIMAXautoregressive integrating moving average model with exogenous inputASM1activated sludge model no. 1BOT-PCTpressure-compensated bottom temperatureCARMAcontrolled autoregressive moving average modelCARIMAcontrolled autoregressive integrating moving average modelCPMcontrol performance monitoringCSTRcontinuous stirred tank reactorCVcontrolled variableDCSdistributed control systemDVdisturbance variableFBPfinal boiling pointFIRfinite impulse responseFSRfinite step responseGPCgeneralized predictive controlIMCinternal model controlLDlinear dynamicLRPIlong-range predictive identificationLSleast squaresLPVlinear parameter varyingLVlimited variableMIMOmulti-input, multi-outputMPCmodel-based predictive controlMSPImulti-step-ahead predictive identificationMVmanipulated variableNARMAXnonlinear autoregressive moving average model with exogenous inputNSnonlinear staticPCTpressure-compensated temperaturePFCpredictive functional controlPIproportional plus integralPIDproportional plus integral plus derivativePLCprogrammable logic controllerPRBSpseudo-random binary signalPRMSpseudo-random multi-level signalPRTSpseudo-random three-level signalPWApiecewise affineRGArelative gain arrayRSTRST polynomial formSISOsingle input, single outputTITOtwo input, two outputTOP-PCTpressure-compensated top temperatureWWTPwastewater treatment plant