Lecture notes on Advanced Control systems

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Lecture notes for the course Advanced Control of Industrial Processes Morten Hovd Institutt for Teknisk Kybernetikk, NTNU November 3, 20092Contents 1 Introduction 9 1.1 Scope of note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Why is process control needed? . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 What knowledge does a process control engineer need? . . . . 12 1.3 The structure of control systems in the process industries. . . . . . . 14 1.3.1 Overall structure . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Mathematical and control theory basics 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Models for dynamical systems . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Dynamical systems in continuous time . . . . . . . . . . . . . 19 2.2.2 Dynamical systems in discrete time . . . . . . . . . . . . . . . 20 2.2.3 Linear models and linearization . . . . . . . . . . . . . . . . . 21 2.2.4 Converting between continuous- and discrete-time models . . . 25 2.2.5 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.6 Similarity transformations . . . . . . . . . . . . . . . . . . . . 28 2.2.7 Minimal representation . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Analyzing linear dynamical systems . . . . . . . . . . . . . . . . . . . 30 2.3.1 Poles and zeros of transfer functions . . . . . . . . . . . . . . 30 2.3.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.3 Frequency analysis . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.4 Bode diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.5 Assessing closed loop stability using the open loop frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 Limitations on achievable control performance 43 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Closed loop transfer functions and closed loop system responses . . . 44 3.4 Limitations on achievable performance . . . . . . . . . . . . . . . . . 45 3.4.1 Control performance in di®erent frequency ranges . . . . . . . 45 3.4.2 Zeros in the right half plane . . . . . . . . . . . . . . . . . . . 46 34 CONTENTS 3.4.3 Unstable systems . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4.4 Time delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.5 Limitations due to uncertainty in the plant model . . . . . . . 51 3.4.6 Limitations due to input constraints . . . . . . . . . . . . . . 52 4 Control structure selection 53 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Common control loop structures for the regulatory control layer . . . 53 4.2.1 Simple feedback loop . . . . . . . . . . . . . . . . . . . . . . . 54 4.2.2 Feedforward control . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2.3 Ratio control . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2.4 Cascade control . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2.5 Auctioneering control . . . . . . . . . . . . . . . . . . . . . . . 57 4.2.6 Split range control . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.7 Parallel control . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.8 Selective control . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.9 Combining basic single-loop control structures . . . . . . . . . 61 4.2.10 Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 Control con¯guration elements and decentralized controller tuning . 64 4.3.1 The relative gain array . . . . . . . . . . . . . . . . . . . . . . 64 4.3.2 The RGA as a general analysis tool . . . . . . . . . . . . . . . 65 4.3.3 The RGA and stability . . . . . . . . . . . . . . . . . . . . . . 67 4.3.4 Summary of RGA-based input-output pairing . . . . . . . . . 69 4.3.5 Alternative interaction measures . . . . . . . . . . . . . . . . . 69 4.3.6 Input-output pairing for stabilization . . . . . . . . . . . . . . 69 4.4 Tuning of decentralized controllers. . . . . . . . . . . . . . . . . . . . 69 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4.2 Loop shaping basics . . . . . . . . . . . . . . . . . . . . . . . 70 4.4.3 Tuning of single-loop controllers . . . . . . . . . . . . . . . . . 70 4.4.4 Multiloop controller tuning. . . . . . . . . . . . . . . . . . . . 75 4.4.5 Tools for multivariable loop-shaping . . . . . . . . . . . . . . . 79 5 Control structure selection and plantwide control 87 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2 General approach and problem decomposition . . . . . . . . . . . . . 88 5.2.1 Top-down analysis . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2.2 Bottom-up design . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Regulatory control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4 Determining degrees of freedom . . . . . . . . . . . . . . . . . . . . . 94 5.5 Selection of controlled variables . . . . . . . . . . . . . . . . . . . . . 95 5.5.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 96 5.5.2 Selecting controlled variables by direct evaluation of loss . . . 98 5.5.3 Controlled variable selection based on local analysis . . . . . . 98CONTENTS 5 5.5.4 An exact local method for controlled variable selection . . . . 101 5.5.5 Measurement combinations as controlled variables . . . . . . . 103 5.5.6 The validity of the local analysis for controlled variable selection104 5.6 Selection of manipulated variables . . . . . . . . . . . . . . . . . . . 105 5.7 Selection of measurements . . . . . . . . . . . . . . . . . . . . . . . . 108 5.8 Mass balance control and throughput manipulation . . . . . . . . . . 109 5.8.1 Consistency of inventory control . . . . . . . . . . . . . . . . . 111 6 Model-based predictive control 115 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Formulation of a QP problem for MPC . . . . . . . . . . . . . . . . . 116 6.2.1 Future states as optimization variables . . . . . . . . . . . . . 120 6.2.2 Using the model equation to substitute for the plant states . . 121 6.2.3 Optimizing deviations from linear state feedback . . . . . . . . 122 6.2.4 Constraints from time n to n+j . . . . . . . . . . . . . . . . 123 6.2.5 Required value for j . . . . . . . . . . . . . . . . . . . . . . . 124 6.2.6 Feasible region and prediction horizon . . . . . . . . . . . . . 125 6.3 Step response models . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.4 Updating the process model . . . . . . . . . . . . . . . . . . . . . . . 126 6.4.1 Bias update . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.4.2 Kalman ¯lter and Extended Kalman Filters . . . . . . . . . . 127 6.4.3 Unscented Kalman ¯lter . . . . . . . . . . . . . . . . . . . . . 130 6.4.4 Receding Horizon Estimation . . . . . . . . . . . . . . . . . . 133 6.4.5 Concluding comments on state estimation . . . . . . . . . . . 139 6.5 Disturbance handling and o®set-free control . . . . . . . . . . . . . . 140 6.5.1 Feedforward from measured disturbances . . . . . . . . . . . . 140 6.5.2 Disturbance estimation and o®set-free control . . . . . . . . . 141 6.6 Feasibility and constraint handling . . . . . . . . . . . . . . . . . . . 142 6.7 Closed loop stability with MPC controllers . . . . . . . . . . . . . . . 144 6.8 Target calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.9 Robustness of MPC controllers . . . . . . . . . . . . . . . . . . . . . 150 6.10 Using rigorous process models in MPC . . . . . . . . . . . . . . . . . 152 7 Some practical issues in controller implementation 155 7.1 Discrete time implementation . . . . . . . . . . . . . . . . . . . . . . 155 7.1.1 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.1.2 Sampling interval . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.1.3 Execution order . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.2 Pure integrators in parallel . . . . . . . . . . . . . . . . . . . . . . . . 157 7.3 Anti-windup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.3.1 Simple PI control anti-windup . . . . . . . . . . . . . . . . . . 159 7.3.2 Velocity form of PI controllers . . . . . . . . . . . . . . . . . . 1606 CONTENTS 7.3.3 Anti-windup in cascaded control systems . . . . . . . . . . . . 160 7.3.4 Hanus' self-conditioned form . . . . . . . . . . . . . . . . . . . 161 7.3.5 Anti-windup in observer-based controllers . . . . . . . . . . . 162 7.3.6 Decoupling and input constraints . . . . . . . . . . . . . . . . 164 7.4 Bumpless transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.4.1 Switching between manual and automatic operation . . . . . . 165 7.4.2 Changing controller parameters . . . . . . . . . . . . . . . . . 166 7.4.3 Switching between di®erent controllers . . . . . . . . . . . . . 166 8 Controller Performance Monitoring and Diagnosis 169 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.2 Detection of oscillating control loops . . . . . . . . . . . . . . . . . . 171 8.2.1 The autocorrelation function . . . . . . . . . . . . . . . . . . . 172 8.2.2 The power spectrum . . . . . . . . . . . . . . . . . . . . . . . 172 8.2.3 The method of Miao and Seborg. . . . . . . . . . . . . . . . . 172 8.2.4 The method of HÄagglund . . . . . . . . . . . . . . . . . . . . . 173 8.2.5 The regularity index . . . . . . . . . . . . . . . . . . . . . . . 174 8.2.6 The method of Forsman and Stattin . . . . . . . . . . . . . . 175 8.2.7 Pre-¯ltering data . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.3 Oscillation diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.3.1 Manual oscillation diagnosis . . . . . . . . . . . . . . . . . . . 177 8.3.2 Detecting and diagnosing valve stiction . . . . . . . . . . . . . 178 8.3.3 Detection of backlash . . . . . . . . . . . . . . . . . . . . . . . 182 8.3.4 Detecting and diagnosing other non-linearities . . . . . . . . . 183 8.4 Root-cause analysis for distributed oscillations . . . . . . . . . . . . . 184 8.5 Control loop performance monitoring . . . . . . . . . . . . . . . . . . 185 8.5.1 The Harris Index . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.5.2 Obtaining the impulse response model . . . . . . . . . . . . . 186 8.5.3 Calculating the Harris index . . . . . . . . . . . . . . . . . . . 188 8.5.4 Estimating the deadtime . . . . . . . . . . . . . . . . . . . . . 188 8.5.5 Modi¯cations to the Harris index . . . . . . . . . . . . . . . . 189 8.5.6 Assessing feedforward control . . . . . . . . . . . . . . . . . . 190 8.5.7 Comments on the use of the Harris index . . . . . . . . . . . . 191 8.5.8 Performance monitoring for PI controllers . . . . . . . . . . . 192 8.5.9 Performance monitoring for cascaded control loops . . . . . . 193 8.6 Multivariable control performance monitoring . . . . . . . . . . . . . 193 8.6.1 Assessing feedforward control in multivariable control . . . . . 194 8.6.2 Performance monitoring for MPC controllers . . . . . . . . . . 194 8.7 Some issues in the implementation of Control Performance Monitoring 195 8.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196CONTENTS 7 9 Linear regression techniques applied in process control 199 9.1 Data pre-treatment and organization . . . . . . . . . . . . . . . . . . 199 9.2 Ordinary Multivariable Linear Regression . . . . . . . . . . . . . . . . 200 9.3 Principal Component Regression. . . . . . . . . . . . . . . . . . . . . 201 9.4 Partial Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 9.5 Detecting anomalies using Principal Component Analysis . . . . . . . 203 A1 Fourier-Motzkin elimination . . . . . . . . . . . . . . . . . . . . . . . 205 A2 Removal of redundant constraints . . . . . . . . . . . . . . . . . . . . 207 A3 The Singular Value Decomposition . . . . . . . . . . . . . . . . . . . 2088 CONTENTSChapter 1 Introduction 1.1 Scope of note This note originates from course notes for the course 'Design og vedlikehold av regu- leringsfunksjoner',given in cooperation between Cyberlab.Org AS and the Engineer- ing Cybernetics Department of the Norwegian University of Science and Technology (NTNU). Parts of this note has later been used in the course Advanced Process Control, which has been o®ered by the Engineering Cybernetics Department in co- operation with the Chemical Engineering Department at NTNU. The most recent version is further adapted for the course Advanced Control of Industrial Processes, o®ered by the Engineering Cybernetics Department. Thetargetaudienceisstudentsinthefourthyearofthe5-yearMScprogrammein Engineering Cybernetics. Thus, the note is written for people with a relatively broad backgroundincontrolengineering,whoarefamiliarwithbothfrequencyresponseand time domain analysis. Whereas frequency response (or Laplace domain) analysis is usedpredominantlyforsingle-loopcontrol,timedomaindescription(indiscretetime) is used extensively in the description of multivariable Model Predictive Control. Concepts from systems theory such as (state) controllability and (state) observ- 1 ability are also used without introduction . It is this authors intent to keep the focus on issues of importance for industrial applications. Frequently, results are presented and discussed, without presenting formal proofs. Readers interested in mathematical proofs will have to consult the references. Readers are also assumed to be familiar with ¯nite dimensional linear algebra, i.e., have a working knowledge of matrices and vectors. Although the subject matter is by necessity of a mathematical nature, mathematical elegance is often sacri¯ced for clarity. In addition to students of control engineering, students with a Process Systems Engineering specialization within Chemical Engineering should also be able to read and bene¯t from this note. 1 Although the importance of these concepts are not exaggerated in this work. 910 CHAPTER 1. INTRODUCTION 1.2 Why is process control needed? Many texts on process control implicitly assume that it is obvious when and why control is needed. It seems obvious that even a moderately complex process plant will be very di±cult to operate without the aid of process control. Nevertheless, it can be worthwhile to spend a few minutes thought on why process control is needed. Inthefollowing,ashortandprobablyincompletelistofreasonsfortheneedofprocess control is provided, but the list should illustrate the importance of process control in a process plant. 1. Stabilizing the process. Many processes have integrating or unstable modes. Thesehavetobestabilizedbyfeedbackcontrol,otherwisetheplantwill(sooner or later) drift into unacceptable operating conditions. In the vast majority of 2 cases, this stabilization is provided by automatic feedback control . Note that in practice, "feedback stabilization" of some process variable may be necessary even though the variable in question is asymptotically stable according to the controlengineeringde¯nitionofstability. Thishappenswheneverdisturbances have su±ciently large e®ect on a process variable to cause unacceptably large variations in the process variable value. Plant operators therefore often use the term "stability" in a much less exact way than how the term is de¯ned in control engineering. A control engineer may very well be told that e.g., "this temperatureisnotsu±cientlystable",eventhoughthetemperatureinquestion is asymptotically stable. 2. Regularity. Evenifaprocessisstable,controlisneededtoavoidshutdownsdue to unacceptable operating conditions. Such shutdowns may be initiated auto- maticallybyashutdownsystem, butmayalsobecausedbyoutrightequipment failure. 3. Minimizing e®ects on the environment. In addition to maintaining safe and stableproduction,thecontrolsystemshouldalsoensurethatanyharmfule®ects ontheenvironmentareminimized. Thisisdonebyoptimizingtheconversionof 3 raw materials , and by maintaining conditions which minimize the production of any harmful by-products. 4. Obtaining the right product quality. Control is often needed both for achieving the right product quality, and for reducing quality variations. 5. Achieving the right production rate. Controlisusedforachievingtherightpro- duction rate in a plant. Ideally, it should be possible to adjust the production rate at one point in the process, and the control system should automatically adjust the throughput of up- or downstream units accordingly. 2 However, some industries still use very large bu®er tanks between di®erent sections in the process. For such tanks it may be su±cient with infrequent operator intervention to stop the bu®er tank from over¯lling or emptying. 3 Optimizing the conversion of raw materials usually means maximizing the conversion, unless this causes unacceptably high production of undesired by-products, or requires large energy inputs.1.2. WHY IS PROCESS CONTROL NEEDED? 11 6. Optimizeprocessoperation. Whenaprocessachievessafeandstableoperation, with little down-time, and produces the right quality of product at the desired production rate, the next task is to optimize the production. The objective of the optimization is normally to achieve the most cost-e®ective production. This involves identifying, tracking and maintaining the optimal operating con- ditions in the face of disturbances in production rate, raw material composition and ambient conditions(e.g., atmospheric temperature). Process optimization often involves close coordination of several process units, and operation close to process constraints. The list above should illustrate that process control is vital for the operation of process plants. Even plants of quite moderate complexity would be virtually impossible to operate without process control. Even where totally manual operation is physically feasible, it is unlikely to be economically feasible due to product quality variationsand highpersonnel costs, since ahigh number ofoperators will be required to perform the many (often tedious) tasks that the process control system normally handles. Usually many more variables are controlled than what is directly implied by the list above, there are often control loops for variables which have no speci¯cation associated with them. There are often good reasons for such control loops - two possible reasons are 1. To stop disturbances from propagating downstream. Even when there are no direct speci¯cation on a process variable, variations in the process variable may cause variations in more important variables downstream. In such cases, it makes sense to remove the disturbance at its source. 2. Local removal of uncertainty. By measuring and controlling a process variable, it may be possible to reduce the e®ect of uncertainty with respect to equip- ment behaviour or disturbances. Examples of such control loops are valve positioners used to minimize the e®ect of valve stiction, or local °ow control loops which may be used to counteract the e®ects of pressure disturbances up- or downstream of a valve, changes in °uid properties, or inaccuracies in the valve characteristics.12 CHAPTER 1. INTRODUCTION 1.2.1 Whatknowledgedoesaprocesscontrolengineerneed? The list on page 10 also indicates what kind of knowledge is required for a process control engineer. The process control engineer needs to have a thorough understand- ing of the process. Most stabilizing control loops involve only one process unit (e.g., a tank ora reactor), and most equipment limitations are also determined bythe indi- vidual units. Process understanding on the scale of the individual units is therefore required. Understanding what phenomena a®ect product quality also require an un- derstanding of the individual process units. On the other hand, ensuring that the speci¯ed production rate propagates throughout the plant, how the e®ect of distur- bances propagate, and optimizing the process operation, require an understanding of how the di®erent process units interact, i.e., an understanding of the process on a larger scale. Mostbasiccontrolfunctionsareperformedbysingleloops, i.e., controlloopswith one controlled variable and one manipulated variable. Thus, when it is understood why a particular process variable needs to be controlled, and what manipulated vari- 4 able should be used to control it , the controller design itself can be performed using traditional single-loop control theory (if any theoretical considerations are made at all). Often a standard type of controller, such as a PID controller, is tuned on-line, and there is little need for a process model. Other control tasks are multivariable in nature, either because it is necessary to resolve interactions between di®erent con- trolloops, orbecausethecontroltaskrequirescoordinationbetweendi®erentprocess units. Process models are often very useful for these types of control problem. The models may either be linear models obtained from experiments on the plant, or possibly non-linear models derived from physical and chemical principles. Some understanding of mathematical modelling and system identi¯cation techniques are then required. Non-linear system identi¯cation from plant experiments are not in standard use in the process industries. Optimizing process operation requires some understanding of plant economics, involving the costs of raw materials and utilities, the e®ect of product quality on product price, the cost of reprocessing o®-spec product, etc. Although it is rare that 5 economicsisoptimizedbyfeedbackcontrollers , anunderstandingofplanteconomics will help understanding where e®orts to improve control should be focused, and will help when discussing the need for improved control with plant management. A process control engineer must thus have knowledge both of process and control engineering. However, it is not reasonable to expect the same level of expertise in either of these disciplines from the process control engineer as for "specialist" process 4 Determining what variables are to be controlled, what manipulated variables should be used for control, and the structure of interconnections between manipulated and controlled variables, are quite critical tasks in the design of a process control system. This part of the controller design is often not described in textbooks on "pure" control engineering, but will be covered in some detail in later sections. 5 It is more common that economic criteria are used in the problem formulation for socalled Real Time Optimization (RTO) problems, or for plant production planning and scheduling.1.2. WHY IS PROCESS CONTROL NEEDED? 13 or control engineers. There appears to be a "cultural gap" between process and control engineers, and the process control engineer should attempt to bridge this gap. This means that the process control engineer should be able to communicate meaningfully with both process and control engineers, and thereby also be able to obtain any missing knowledge by discussing with the "specialists". However, at a production plant there will seldom be specialists in control theory, but there will always be process engineers. At best, large companies may have control theory specialists at some central research or engineering division. This indicates that a process control engineer should have a fairly comprehensive background in control engineering, while the process engineering background should at least be su±cient to communicate e®ectively with the process engineers. Inthesamewayasforotherbranchesofengineering,successatworkwillnotcome from technological competence alone. A successful engineer will need the ability to work e®ectively in multi-disciplinary project teams, as well skills in communicating with management and operators. Such non-technical issues will not be discussed further here.14 CHAPTER 1. INTRODUCTION 1.3 Thestructureofcontrolsystemsintheprocess industries. 1.3.1 Overall structure When studying control systems in the process industries, one may observe that they often share a common structure. This structure is illustrated in Fig. 1.1. Production planning/ scheduling Real time optimization Supervisory control Regulatory control To manipulated variables From measurements Process Figure 1.1: Typical structure of the control system for a large plant in the process industries. ThelowerlevelinthecontrolsystemistheRegulatorycontrol layer. Thestructure of the individual controllers in the regulatory control layer is normally very simple. Standard single-loop controllers, typically of PI/PID type are the most common, but other simple control functions like feed forward control, ratio control, or cascaded1.3. THESTRUCTUREOFCONTROLSYSTEMSINTHEPROCESSINDUSTRIES.15 control loops may also be found. Truly multivariable controllers are rate at this level. The regulatory control system typically controls basic process variables such as temperatures, pressures, °owrates, speeds or concentrations, but in some cases the controlled variable may be calculated based on several measurements, e.g., a component°owratebasedonmeasurementsofbothconcentrationandoverall°owrate or a ratio of two °owrates. Usually a controller in the regulatory control layer manipulates a process variable directly (e.g., a valve opening), but in some cases the manipulated variable may be a setpoint of a cascaded control loop. Most control functions that are essential to the stability and integrity of the process are executed in this layer, such as stabilizing the process and maintaining acceptable equipment operating conditions. The Supervisory control layer coordinates the control of a process unit or a few closely connected process units. It coordinates the action of several control loops, and tries to maintain the process conditions close to the optimal while ensuring that operating constraints are not violated. The variables that are controlled by super- visory controllers may be process measurements, variables calculated or estimated from process measurements, or the output from a regulatory controller. The ma- nipulated variables are often setpoints to regulatory controllers, but process variables may also be manipulated directly. Whereas regulatory controllers are often designed and implemented without ever formulating any process model explicitly, supervisory controllers usually contain an explicitly formulated process model. The model is dynamic and often linear, and obtained from experiments on the plant. Typically, supervisory controllers use some variant of Model Predictive Control (MPC). The optimal conditions that the supervisory controllers try to maintain, may be calculated by a Real Time Optimization (RTO) control layer. The RTO layer identi¯estheoptimalconditionsbysolvinganoptimizationprobleminvolvingmodels of the production cost, value of product (possibly dependent on quality), and the process itself. The process model is often non-linear and derived from fundamental physical and chemical relationships, but they are usually static. ThehighercontrollevelshowninFig. 1.1isthe Production planning and schedul- ing layer. This layer determines what products should be produced and when they shouldbeproduced. Thislayerrequiresinformationfromthesalesdepartmentabout the quantities of the di®erent products that should be produced, the deadlines for delivery, and possibly product prices. From the purchasing department information about the availability and price of raw materials are obtained. Information from the plant describes what products can be made in the di®erent operating modes, and what production rates can be achieved. In addition to the layers in Fig. 1.1, there should also be a separate safety system that will shut the process down in a safe and controlled manner when potentially dangerous conditions occur. There are also higher levels of decision making which are not shown, such as sales and purchasing, construction of new plants, etc. These levels are considered to be of little relevance to process control, and will not be discussed further. Note that there is a di®erence in time scale of execution for the di®erent lay- ers. The regulatory control system typically have sampling intervals on the scale of16 CHAPTER 1. INTRODUCTION one second (or faster for some types of equipment), supervisory controllers usually operateonthetimescaleofminutes, theRTOlayeronascaleofhours, andtheplan- ning/schedulinglayeronascaleofdays(orweeks). Thecontrolbandwidthsachieved by the di®erent layers di®er in the same way as sampling intervals di®er. This di®er- ence in control bandwidths can simplify the required modelling in the higher levels; if a variable is controlled by the regulatory control layer, and the bandwidth for the control loop is well beyond what is achieved in the supervisory control layer, a static model for this variable (usually the model would simply be variable value = setpoint) will often su±ce for the supervisory control. It is not meaningful to say that one layer is more important than another, since they are interdependent. The objective of the lower layers are not well de¯ned without information from the higher layers (e.g., the regulatory control layer needs to know the setpoints that are determined by the supervisory control layer), whereas the higher layers need the lower layers to implement the control actions. However, in many plants human operators perform the tasks of some the layers shown in Fig. 1.1, it is only the regulatory control layer that is present (and highly automated) in virtually all industrial plants. Why has this multi-layered structure for industrial control systems evolved? It is clear that this structure imposes limitations in achievable control performance comparedtoahypotheticaloptimalcentralizedcontrollerwhichperfectlycoordinates all available manipulated variables in order to achieve the control objectives. In the past, the lack of computing power would have made such a centralized controller virtually impossible to implement, but the continued increase in available computing power could make such a controller feasible in the not too distant future. Is this the direction industrial control systems are heading? This appears not to be the case. In the last two of decades development has instead moved in the opposite direction, as increased availability of computing power has made the Supervisory control and Real Time Optimization layers much more common. Some reasons for using such a multi-layered structure are: ² Economics. Optimal control performance - de¯ned in normal control engi- neering terms (using e.g., the H ¡ or H norm) - does not necessarily imply 2 1 optimal economic performance. To be more speci¯c, an optimal controller synthesis problem does not take into account the cost of developing and main- taining the required process (or possibly plant economic) models. An optimal centralized controller would require a dynamic model of most aspects of the process behaviour. The required model would therefore be quite complex, and di±cult to develop and maintain. In contrast, the higher layers in a structured control system can take advantage of the model simpli¯cations made possible by the presence of the lower layers. The regulatory control level needs little model information to operate, since it derives most process information from 6 feedback from process measurements . 6 A good process model may be of good use when designing control structures for regulatory1.3. THESTRUCTUREOFCONTROLSYSTEMSINTHEPROCESSINDUSTRIES.17 ² Redesign and retuning. The behaviour of a process plant changes with time, for a number of reasons such as equipment wear, changes in raw materials, changes in operating conditions in order to change product qualities or what productsareproduced,andplantmodi¯cations. Duetothesheercomplexityof a centralized controller, it would be di±cult and time-consuming to update the controller to account for all such changes. With a structured control system, it is easier to see what modi¯cations need to be made, and the modi¯cations themselves will normally be less involved. ² Start-up and shutdown. Common operating practice during start-up is that many of the controls are put in manual. Parts of the regulatory control layer may be in automatic, but rarely will any higher layer controls be in operation. The loops of the regulatory control layer that are initially in manual are put in automatic when the equipment that they control are approaching normal operating conditions. When the regulatory control layer for a process section isinservice,thesupervisorycontrolsystemmaybeputinoperation, andsoon. Shutdown is performed in the reverse sequence. Thus, there may be scope for signi¯cant improvement of the start-up and shutdown procedures of a plant, as quicker start-up and shutdown can reduce plant downtime. However, a model which in addition to normal operating conditions also is able to describe start- up and shutdown, is necessarily much more complex than a model which covers onlytherangeofconditionsthatareencounteredinnormaloperation. Building such a model would be di±cult and costly. Start-up and shutdown of a plant with an optimal centralized control system which does not cover start-up and shutdown, may well be more di±cult than with a traditional control system, because it may not be di±cult to put an optimal control system gradually into or out of service. ² Operator acceptance and understanding. Controlsystemsthatarenotaccepted by the operators are likely to be taken out of service. An optimal centralized control system will often be complex and di±cult to understand. Operator understandingobviouslymakesacceptanceeasier, andatraditionalcontrolsys- tem, being easier to understand, often has an advantage in this respect. Plant shutdowns may be caused by operators with insu±cient understanding of the control system. Such shutdowns should actually be blamed on the control system (or the people who designed and installed the control system), since operators are an integral part of the plant operation, and their understanding of the control system must therefore be ensured. ² Failure of computer hardware and software. In traditional control systems the operators retain the help of the regulatory control system in keeping the process in operation if a hardware or software failure occurs in higher levels of the control system. A hardware backup for the regulatory control system is control. However, after the regulatory controllers are implemented, they normally do not make any explicit use of a process model.18 CHAPTER 1. INTRODUCTION much cheaper than for the higher levels in the control system, as the regulatory control system can be decomposed into simple control tasks (mainly single loops). In contrast, an optimal centralized controller would require a powerful computeranditisthereforemorecostlytoprovideabackupsystem. However, with the continued decrease in computer cost this argument may weaken. ² Robustness. The complexity of an optimal centralized control system will make it di±cult to analyze whether the system is robust with respect to model uncertainty and numerical inaccuracies. Analyzing robustness need not be trivial even for traditional control systems. The ultimate test of robustness will be in the operation of the plant. A traditional control system may be applied gradually, ¯rst the regulatory control system, then section by section of the supervisory control system, etc. When problem arise, it will therefore be easier to analyze the cause of the problem with a traditional control system than with a centralized control system. ² Local removal of uncertainty. It has been noted earlier that one e®ect of the lower layer control functions is to remove model uncertainty as seen from the higher layers. Thus, the existence of the lower layers allow for simpler models in the higher layers, and make the models more accurate. The more complex computations in the higher layers are therefore performed by simpler, yet more accurate models. A centralized control system will not have this advantage. ² Existing traditional control systems. Where existing control systems perform reasonably well, it makes sense to put e®ort into improving the existing system rather than to take the risky decision to design a new control system. This argument applies also to many new plants, as many chemical processes are not well understood. For such processes it will therefore be necessary to carry out model identi¯cation and validation on the actual process. During this period some minimum amount of control will be needed. The regulatory control layer ofatraditionalcontrolsystemrequireslittleinformationabouttheprocess,and can therefore be in operation in this period. It should be clear from the above that this author believes that control systems in thefuturewillcontinuetohaveanumberofdistinctlayers. Twoprerequisitesappear to be necessary for a traditional control system to be replaced with a centralized one: 1. The traditional control system must give unacceptable performance. 2. Theprocessmustbesu±cientlywellunderstoodtobeabletodevelopaprocess model which describes all relevant process behaviour. Sinceitisquiterarethatatraditionalcontrolsystemisunabletocontrolaprocess for which detailed process understanding is available (provided su±cient e®ort and expertise have been put into the design of the control system), it should follow that majority of control systems will continue to be of the traditional structured type.Chapter 2 Mathematical and control theory basics 2.1 Introduction This section will review some mathematical and control theory basics, that in actual fact is assumed covered by previous control courses. Both the coverage of topics and their presentation will therefore be sketchy and incomplete, aimed at ² correcting what is this author's impression of what are the most common mis- conceptions among students who follow this course, as well as ² to establish some basic concepts and introduce some notation. 2.2 Models for dynamical systems Many di®erent model representations are used for dynamical systems, and a few of the more common ones will be introduced here. 2.2.1 Dynamical systems in continuous time A rather general way of representing a dynamical system in continuous time is via a set of ordinary di®erential equations: x_ =f(x;u;d) (2.1) dx wherethevariablesxaretermedthesystemstatesandx_ = isthetimederivativeof dt thestate. Thevariablesuanddarebothexternalvariablesthata®ectthesystem. In the context of control, it is common to distinguish between the manipulated variables or (control) inputs u that can be manipulated by a controller, and the disturbances d that are external variables that a®ect the system but which cannot be set by the controller. 1920 CHAPTER 2. MATHEMATICAL AND CONTROL THEORY BASICS The system states x are generally only a set of variables that are used to describe the system's behaviour over time. Whether the individual components of the state vector can be assigned any particular physical interpretation will depend on how the model is derived. For models derived from fundamental physical and chemical relationships (often termed 'rigorous models'), the states will often be quantities like temperatures, concentrations, velocities, etc. If, on the other hand, the model is an empirical model identi¯ed from observed data, it will often not be possible to assign any particular interpretation to the states. Along with the state equation (2.1), one typically also needs a measurement equa- tion such as y =g(x;u;d) (2.2) where the vector y is a vector of system outputs, which often correspond to available physical measurements from the systems. Control design is usually at its most simple when all states can be measured, i.e., when y =x. Disturbances need not be included in all control problems. If no disturbances are included in the problem formulation, equations (2.1) and (2.2) trivially simplify to x_ =f(x;u) and y =g(x;u), respectively. Since we are dealing with dynamical systems, it is hopefully obvious that the variables x;y;u;d may all vary with time t. In this section time is considered as a continuous variable - in accordance with our usual notion of time. Together, equations (2.1) and (2.2) de¯ne a system model in continuous time. This type of model is rather general, and can deal with any system where it su±ces to consider system properties at speci¯c points in space, or where it is acceptable to average/lump system properties over space. Such models where properties are averaged over space are often called lumped models. For some applications, it may be necessary to consider also spatial distribution of properties. Rigorous modelling of such systems typically result with a set of partial di®erential equations (instead of the ordinary di®erential equations of (2.1)). In additiontoderivativeswithrespecttotime,suchmodelsalsocontainderivativeswith respect to one or more spatial dimensions. Models described by partial di®erential equations will not be considered any further in these notes. Although control design based on partial di®erential equations is an active research area (in the area of °ow control, in particular), the more common industrial practice is to convert the set of partialdi®erentialequationstoa(larger)setofordinarydi®erentialequationsthrough some sort of spatial discretization. 2.2.2 Dynamical systems in discrete time Although time in the 'real world' as we know it is a continuous variable, control systems are typically implemented in computer systems, which cyclically execute a set of instructions. Measurements and control actions are therefore executed at discrete points in time, and to describe system progression from one time instant to subsequent instants we will need a discrete time model. Such models may be

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