Dynamic Systems and Control Technology

Transcription

1 Dynamic Systems and Control Technology Heinz A. Preisig Systems & Control Group Technical Physics W&S 1.32 Eindhoven University of Technology 5600 MB Eindhoven The Netherlands Chapter 1-9: version 1 March 1996 Chapter 0,1,2,3: version 2, September 1998

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3 Contents Some Words in Advance: 0.1 Heinz A. Preisig i 03/1999

4 Some Words in Advance: Control, which is the second of the two main subjects, builds on the knowledge of the dynamics of the plant simply because the only purpose of control is to impose a certain dynamic behaviour on the plant by the means of controlling the plant s interaction with its environment. Modelling and the study of dynamics characteristics is thus a prerequisite. It is then also the first part of the course, which is a rather large piece. Since we do not deal with a particular plant, but have the objective to build our knowledge such that it can be applied to almost any plant, we have to undergo a step in the level of abstraction. Because controlling means complete knowledge it requires abstraction of everything. Control ties together very many of the subjects that are offered in a standard curriculum of an engineer, namely the basic principles of physics such as the conservation laws, but also transfer laws for energy, mass, momentum, and all kind of work forms. Control needs to know it all. It is one of the first and often only course that challenges the student on this width of knowledge and his depth of knowledge in his profession. This lecture is not a book. It is a concentrate of knowledge taken from many different sources. The original objective was to reduce the need for taking notes during the lecture. But, whilst the notes have become more extensive, taking notes is still necessary as the manuscript is not complete. Having a script available makes it also possible for the lecturer to follow a different path during the presentation than is followed in the script. The idea behind such an approach is that the student gets two views and at least two experiences, namely the lecturer s presentation and the material as it is described in the script. The third pass is done in the exercises and a fourth in projects or laboratory assignments, if so applicable to the course. The last pass should be your preparation of the examination culminating in your display of mastery of the subject. The script contains more material than is discussed during the lecture. For a beginner s course, the discussion on stability is minimised to the eigenvalue/pole argument. The canonical representations are only discussed on the surface, with the exception of the diagonalised system representation. Observability is include in the discussion in its full depth, whilst controllability again only as a principle. The discussion of the derivation is minimised on the latter issue. The main emphasis in control is placed on PID, that is fixed-structure controllers based on the argument that they represent the most common type of controllers in industrial applications after the on-off controllers. It was felt that an introduction to discrete systems should be included because of the flood of computer-controlled systems also in a laboratory environment. Laboratory automation is indeed an objective that may be covered by a course based on this material. If time permits, the beginner s course should include the concepts of model predictive control. In the past at least the concept of observers was included at the very end of the program. This course asks the student to climb a step up. Forsomeitisalargestep,forsomeitis 0.1

5 0.2 CONTENTS not so large. In any case it needs some effort but with a bit of helping support, it will work out. If we can be of help please let us know. Also any comments towards the improvement of any part of the course are appreciated and welcomed.

6 Contents 1 Introduction Motivation The Concept Control Control Structures Heinz A. Preisig ii 08/1999

7 Chapter 1 Introduction Synopsis A controller imposes a prescribed behaviour onto the plant. The more the controller is able to anticipate the behaviour of the plant the better it will perform in its task. It must thus know something about the dynamics of the plant, but it must also know how tight it should control the plant that is it must know the expected performance of the controlled plant. 1.1 Motivation Adding a controller to a plant has one and only one goal, namely to modify the dynamic behaviour of the plant. The controller should change the dynamic behaviour to meet the performance conditions one has defined for the plant. Controllers are used to force the plant along a pre-described trajectory. The process is being steered for which reason the design problem is called the steering or servo problem. Processes are not only steered, but controllers are also used to compensate for disturbances, which have there origin in a change of the environment of the plant. Control of a plant can only be done through manipulation of plant-external forces. They are used to move the plant in the desired direction. The controller must thus anticipate what the plant will be doing if an external force is being changed. In physical plants, these external forces may be mass flows, heat flows, or mechanical work to mention the main ones. Take the example of controlling the temperature in a house. The house is equipped with a heating, the heat source, and a temperature measurement and a controller. The controller gets information aboutthe state of thehouse froma temperature measurement located somewhere inside the house. The temperature of the house is affected by all heat sources present in the house and the heat losses or gains due to changes in the environment temperature of the house. A simple on-off controller device, though, will not acquire information about the various heat sources and heat losses but knows only about some temperature in some room of the house. If an indirect heating system is used, where hot water is being circulated through the radiators distributed in the house, additional dynamic effects are observed: The temperature will keep on rising (falling) after the heating has been switched off (on). This continued rise (drop) of the temperature is due to the capacity effects of the water system. Assuming that the switching device has a certain hysteresis, the temperature will not only 1.1

8 1.2 CHAPTER 1. INTRODUCTION vary in the band between the limits of the hysteresis, but it will overshoot and undershoot. An intelligent controller would obviously try to anticipate the behaviour of the plant and switch the heater on or off before the corresponding limit has been reached. Anticipation may not always be sufficient, though. Take the example of controlling the flow and the temperature of water in a shower. Both, flow and temperature are changed by manipulating the hot and the cold water flow by turning the cranks of two facets. Temperature and water flow are sensed by the person taking the shower who converts the sensations into a comfort level which is used to decide on the control action. If the temperature is too low, normally one would open the hot water facet more until the maximum is reached. Only then is the flow of cold water being reduced until the temperature is acceptable. The comfort level is thus moving the process towards a high flow and an acceptable temperature range. Assume now that the flow of hot water changes because a new hot water user in the house is drawing a significant amount from the common hot water source. With the new user getting on-line, the water pressure drops in the pipe and consequently the hot water flow drops and with it the mixing temperature. If the flow of hot water drops dramatically, the person under shower will jump quickly away from the freezing stream, grabbing the cold facet quickly and reducing the cold water flow drastically in an attempt to bring the temperature back up again. However, the temperature change occurs not instantaneously. It takes some time before the water has travelled from the facet through the hose to the shower head down to the person s hand sensing the temperature level, particularly if the water pressure is low and the hose is long. He or she is probably quite impatient and has reduced the cold water flow further before the original change reaches the sensing hand. In addition, the reducing of the cold water flow also changes the total flow and with it the pressure drop in the shower assembly. Consequently, the flow of the hot water increases again. Most likely, the result is an overshoot of the temperature. After a near freeze, the body of the victim is now almost boiled as the hot water wave arrives on his/her skin. Frantically, he/she reaches for the facets, the cold one again and increases the flow. Having experienced a very unpleasant flush of hot water a healthy correction is applied. This, in turn, changes things the other way around again. The consequent change is then experienced again with delay and so forth. If in addition, the new user is also manipulating the flow, taking a shower becomes a very involved control activity. Fortunately, a human will learn about the dynamic behaviour of the process and will generally improve its ability to control the process. He/she is able to adjust to different conditions, in the case of the shower, to different type of showers, different water pressure conditions, etc. A mechanical control device has a very limited or no ability to adapt to such changes. The human, on the other hand, has excellent abilities that let him adapt to a changing situation. It is for this reason that a human being is very often part of the overall control loop in a plant. Control is focusing on improving the performance of the controlled system. Thus all activities that contribute towards this goal are measured in terms of the final performance. Anticipation, for example, requires a model of the process. The model of the process, though, must only be as good as to enable the controller to take an appropriate action. The same applies to the controller. It must only be able to meet the goal. A too good model will usually waste computing resources and a too good controller may result in a too large number of control actions causing unnecessary ware or may cause other long-term problems. The ultimate consequence is that the successful running of a plant is the result of an iterative process which involves learning about the dynamics of the process, designing a controller, testing of the procedure on a simulated process followed by an installation of the controller on the plant. Only then can the performance of the controlled plant be measured. Unsatisfactory

9 1.2. THE CONCEPT CONTROL 1.3 performance will then trigger a next iteration until the performance goals are met or the designer gives up. Understanding the dynamic behaviour of the plant is obviously important for the subject. If the model of the process is wrong, independent of the nature of the model, the anticipation, which is eventually determining the control operation, is similarly wrong. The magnitude of the error may even be amplified by the controller and yield disastrous overall results. Modelling of the dynamic process is thus important and consequently a corresponding emphasis is placed on this subject. The analysis of the process is to a large degree done on the process substitute, namely the model. If the model is a mathematical object, which it usually is in our case, the analysis is a mathematical one. The method of analysis is largely determined by the nature of the mathematical model. A classification and discussion of the mathematical structures and some of their consequences is appropriate. Most of this material is housed in Mathematical System Theory, though some of it is spread over various disciplines such as Signal Processing, Information Theory and Statistics. Before, though, some of these subjects are being entered, followed by the discussion of particular control structures and strategies, the motivating introduction is continued with the discussion of some basic control structures. 1.2 The Concept Control Control Structures One way of controlling a system is to simply dictate what the process has to do. In this case, the controller imposes its commands on the process without having any knowledge of the state of the plant. It simply ignores it. Figure 1.1 shows the arrangement schematically. set point control controlled signal variables C :: Controller P :: Plant plant plant input output Figure 1.1: Open-loop control structure The controller assumes that the process is actually able to do what it commands. If for some reasons this is not the case, it may be the cause of problems. It may be that the environment of the plant changes and with it the effects the environment has on the plant, usually categorised as disturbances. Disturbances can be effects like changing raw product quality, different conditions of the feed, the plant or the environment, such as different environment temperatures, to mention a few. It seems logical, that the overall performance in such a situation can be improved if the controller would know about these problems. This then requires that this information is extracted from the plant by the means of measurements, which then are being made available to the controller. Introducing what has been suggested into the scheme results in a loop structure as shown in 1.2. The scheme leaves a few things open, which for the control engineer may be obvious but may not be so obvious for others. For example, it is not quite clear what is being controller in this plant. Without specifying a particular plant, what one can say is that in general not

10 1.4 CHAPTER 1. INTRODUCTION set point control controlled signal variables C :: Controller P :: Plant plant plant input output measurement of plant conditions Figure 1.2: Generic closed-loop control structure all effects of the environment of the plant are controlled by the controller, but only some. This leaves some environment effects to act freely on the system. The term environment is here used in a generic sense in that anything that is not plant is environment. This may also include the feed sources and the product sinks, for example. If these uncontrolled effects are main streams such as reactant or product streams, energy inflows, etc. one calls them loads. Already the problem of talking about inputs and outputs becomes visible, a subject that needs careful elaboration. set point -1 control C :: Controller error control signal plant input load & disturbances controlled variables P :: Plant Figure 1.3: Closed-loop control structure with load and other environment effects Often, the desired behaviour is expressed in desirable conditions. Preferably these conditions are also measured and compared with the desired settings, the setpoint. The difference is called the control error. Such a scheme establishes a loop as it is shown in 1.3. At this point it is appropriate to take a brief rest and reflect on the two basic schemes and see if once can derive principle properties or relations for such systems. If one is interested in just principle ideas, that is not an analysis that applies to single problems, but to as many as possible, it is appropriate to make quite rigorous assumptions. The choice of assumptions points towards idealised situation. In this case an important result is obtained when one asks for perfect control. y s u C y P y s e C u y K P -1 Figure 1.4: Open-loop and closed-loop control signal flow structure

11 1.2. THE CONCEPT CONTROL 1.5 The ideal overall behaviour of the scheme as shown in 1.4 is that the output follows exactly the setpoint, that is This, in turn, implies that y = y s (1.1) = PCy s (1.2) PC = I or C = P 1 (1.3) The answer to the controller design problem asking for perfect control is that the controller is the inverse of the plant. Given the plant is a physical system, the controller can in the very best case be the inverse of the model of the plant. This provides a very strong motivation for modelling the plant. A similar analysis can be done for the closed-loop system. In this case the equations are In this case the relation holds y = y s (1.4) = (I + PC) 1 PC K y s (1.5) (I + PC) 1 PC K = I or C = P 1 (K I) 1 (1.6) Again, the controller is related to the inverse of the plant. Normally, the control schemes do not show the gain K. In that case, the ideal controller excerpts infinite amplification, as the term K I 0. C ff set point -1 control error C fb control signal plant input load disturbance controlled variables P Figure 1.5: Feedforward control combined with feedback control An interesting variation of these schemes arises when one measures the quantities that have an effect on the plant. If a change in these effects is detected, then one can anticipate their effects on the plant and counteract these effects by the means of a control action in advance. The advantage of this scheme, called the feedforward scheme, is that one does not wait until the disturbance propagates through the plant to the outputs or the positions in the plant where the conditions are being measured, but that the control action is applied earlier, namely when the disturbance is detected at the entry to the plant.

12 1.6 CHAPTER 1. INTRODUCTION There are many more schemes in use. The ones above are just the basic schemes. The world of ideas has no limits. Using a model, for example, is very explicitly done in internal model control, abbreviated with IMC. It runs a model in parallel to the process and looks at the difference between the actual process output and the output predicted by the model. The idea derives from the open-loop scheme. From the analysis of perfect control, we know that the open-loop scheme as it is shown on the left results a controller that is the inverse of the plant. If we now run the model of the plant in parallel of the plant, so it gets the same input and if the model is perfect, that is it describes the plant s input/output behaviour so well that no difference in the output can be detected, then the output error approaches zero. Consequently, in this ideal case, the control loop on the right has the identical behaviour than the one on the left. This scheme appears sensitive to modelling errors, because if the y s u C y P y s e C u y P ỹ -1 M -1 m Figure 1.6: Internal Model Control concept model does not describe the plant well, the output error would typically be non-zero and the controller would try to compensate even when it is not necessary. The way out of this problem is to modify the controller defining it as C := (I + CM) 1 C (1.7) which is called the Internal Model Controller because the model appears explicitly in side the controller. Yet another interesting idea is to play with the different time scales of disturbances and dynamic parts of the plant. In this case it is of advantage to construct a hierarchical control structure. In figure 1.7 the inner loop takes care of the fast process component and the outer loop watches the slower part. The slower loop is always the outer loop. The opposite would y so e o y si e i u P C C o i P f y i -1-1 P s y Figure 1.7: Cascade control structure with separate loops for the fast and the slow part of the plant quite obviously cause problems. This control structure is called cascade control. It is almost always present when the plant has some local control facilities, such as local flow controllers or local speed controllers, to mention just two. Having a model for the plant, that is a description of the dynamic behaviour of the plant in one or the other form, seems very important for good performance. What if the plant changes with time, performance would then deteriorate depending on the changes of the behaviour of the plant. If the changes can be detected, it seems appropriate to think about

13 1.2. THE CONCEPT CONTROL 1.7 adj y s e -1 C slow loop u P y i fast loop y Figure 1.8: Adaptive control with a fast inner control loop and a slow outer adjustment adjusting the controller as the plant changes. This implies though that a number of things must be added. Figure 1.8 shows such a scheme. Again one can identify two loops. The controller can also take the task to stabilise unstable plants, a technique that is today quite frequently used in for example aeroplanes not so in civil aircrafts but frequently in military machines. Control tasks vary with the plant and the performance specifications being imposed. Knowledge of the plant is thereby essential. Modelling thus plays an essential role in today s control and will increasingly do so in the future as model-based control becomes more and more the rule rather than the exception.

14 Contents 2 Modelling of Dynamic Systems in the Time Domain System s Approach The Concept System The Concept Model The Model as a Memory Parameterisation The Modelling Relation The Concept State Modelling: A Design Activity What is Modelling? A Step-Wise Approach to Primary Modelling Primitive Model Components Basic Definitions Example Pseudo Phase Mathematical Description of Primitive Systems Example Flows of extensive quantities and conjugate intensive properties, the potentials Mass Balances Energy Balances Mathematical Description of Connections Reference Co-ordinate System Sign Relations Jump Relations Example Nernst s distribution constant Changing from One State of Aggregation to Another Production Term Heinz A. Preisig iii 08/1999

15 Example Stoichiometry State-Variable Transformations Example Mass Balances and Variable Transformations State-Space Representation Example Inputs and outputs Making Assumptions Model Granularity Example Fixed-Bed Reactor Species Topology : Uni-Directional Flows Primitive Systems Enthalpy Balance for Reactive Systems at Constant Pressure Mechanical Energy Balance Steady-State Assumptions Unmodelled Mass Flows Example Primitive Salt Plant Derivation of Transfer Laws A Theme on Singular Perturbation Model Classification Static - Dynamic Models Lumped System Models Ordinary Differential Equation Models Linearity Time Invariance Linear, Time-constant, Causal, Time-Invariant Systems Distributed System Models Partial Differential Equation Models Steady-State Models Discrete Models Time-Constant Models Stochastic Systems Models Citations Heinz A. Preisig iv 08/1999

16 Chapter 2 Modelling of Dynamic Systems in the Time Domain Synopsis The efforts to successively extend the knowledge of dynamic systems lead to a very general theoretical framework with its own terminology. Particularly, the linguistic terms used in system theory are introduced and discussed from the viewpoint of modern philosophy of science. Though it seems to have derived independently. The framework really derives from macroscopic system s analysis. 2.1 System s Approach The idea of the system approach is very generic. It defines systems as entities being part of the global system usually referred to universe. In applying the system s approach one observes the change of the system and all things that affect the system from the outside The Concept System So what is a system precisely? Webster defines it as a regularly interacting or interdependent group of items forming a unified whole. This definition makes two main points, namely that a system consists of items or components and that these components may interact. We shall use this definition in its essential parts but extend it by allowing for recursion, that is, a system exists of subsystems and each of these subsystems may be a system itself, thus may consist itself of subsystems. This iterative definition allows for a corresponding refinement and thus allows for the adjustment of the model granularity to the detail required by the application. Webster s definition of a system uses the term group, which implies some communal property or attribute of the group members. The term unified whole can be interpreted as the system which is defined by its closed boundary. This definition of system is cyclic tied to the definition of the boundary separating the system from its environment: the system is defined by its boundary or the boundary is defined by the system. In any case, the system and its environment form together the whole of what exists, namely the universe. 2.1

17 2.2 CHAPTER 2. MODELLING OF DYNAMIC SYSTEMS IN THE TIME DOMAIN The term system is so generic, that one could map almost everything into it. Everybody is thus working with systems. There is, however, a branch of science which made systems the central object of its research, namely system theory. This branch of science, which most appropriately is assimilated to applied mathematics, is the result of an evolutionary process transmuting classical physics into cybernetics, classical signal theory, control theory, system theory and finally abstract algebraic system theory. The major characteristic that distinguishes system theory from classical theory is its generality and abstractness; it is concerned with the mathematical properties of the set of equations that describe a system and not the system itself (Zadeh 1962 [24]). The nature of the system is captured in a mathematical description, which is used for the further developments instead of the plant. Abstract system theory is concerned with the interconnection of subsystems and characterising the dynamic behaviour of the overall system in terms of the dynamic behaviour of its constituent subsystems. This is done on abstract objects that simulate the behaviour of the real-world object, the prototype. The description of the prototype in form of a mathematical model is a prerequisite for applying mathematical system theoretical methods. The operation modelling of a prototype is therefore of particular importance since any result of the analysis or any design is dependent on the model of the prototype. If one comes to think about it, then most technical studies are focusing on this question in one or the other way and it is really the major subject throughout the chemical engineering curriculum. It is important to recognise that once a system has been modelled, the analysis of the model and indirectly of the system has been reduced to a pure mathematical problem for which a solution can be sought within the framework of mathematics independent of the original system. The first step, namely the mapping of a physical object, such as a plant, into a mathematical object is critical, as it constraints the information contents of the model. Choosing a model structure puts hard limits on what the model can describe The Concept Model Acquisition of data for the purpose of observing systems and finding and testing theories that map the system into models is at the heart of natural science and engineering. The construction of a model, the theory, is a type of idealisation projecting the main characteristics of the object into a model of the system and is, at the same time, a classification of system by exploring common projection relations called theories. The term model in its original meaning implies a change of scale, the model as a more or less good copy of an object, the prototype, on a different scale. The prototype is usually taken as a physical system and the change in scale is usually be meant as a change in the physical variable length. A very typical example of the term model in this common interpretation is a child s toy, such as a steam engine or a playhouse etc. However, it is also possible to model other than physical systems such as sociological, psychological, economical systems. It might actually be any type of system within the definition of the term system. One can think of different scaling procedures in which other variables are scaled. Scaling depends on the nature of the modelling problem and affects in general any process variable. A precise definition of the term model in this more general context becomes very difficult, a fact that manifests itself in the number of unsuccessful attempts made to define the term model, which are documented in the literature of Philosophy of Science i.e. Swanson 1966 [22], Farre 1967 [9], Aris 1978 [2], Aris 1979 [3], Kammler 1974 [16], Hoerz 1976 [12]. Any of these attempts

18 2.1. SYSTEM S APPROACH 2.3 resulted in a very vague definition comprising as many of the aspects in which the term model is used as possible. It is probably really of no importance to the engineer to know such a definition but it is essential that he understands the idea of modelling which is closely related to the purpose for which models are constructed. Models make it possible to study the behaviour of the system within the domain of common characteristics of the model and the modelled system without affecting the original system. Thus, it is the common part, the homomorphism or the analogy between the prototype and the model and the homomorphism of the different relations between the two that are of interest and which are the two factors characterising the idea model The Model as a Memory In its characteristic as a projection, modelling could be regarded as a data reduction technique. Rather than storing all possible records of output data generated by exciting the system with every possible input record, only the relation between the two data records, the model of the system, must be stored. The models can then be used to reconstruct the output signals given any specific input signal thus simulate the system responses which is an important feature extensively used in analysis and design of dynamic processes. The inverse problem, namely to identify the model, given a set of input and output data, is very fundamental to natural and technical sciences. The product of the identification procedure can be viewed as the remainder after the non-essential parts and/or aspects of the system have been excluded (Eykhoff 1979 [8], Shinnar 1978 [21]) Parameterisation Parameterisation is considered as the next level of abstraction. It splits the memorisation of the transfer behaviour up into two parts, the structure of the model and the parameters. Thus, for different, but structurally equivalent systems only one structure must be stored, supplemented with a file of parameter sets, each characterising a particular system. The model structure together with a particular set of parameters represent a concentrate describing the essential characteristics of the particular system. This is the reason for which the parameters are described as the characteristic values of the model. Parameters are only defined in connection with a particular model structure or in other words, the parameters are defined by the model. As a consequence parameters can only be used in connection with the corresponding model structure. They have no meaning in the context of another, structurally foreign model even in the case where both models are projections of one and the same system using different theories. An assertion which relates the parameters of two theoretically different models, can only be constructed if a relation between the two models can be formulated by relating the two theories The Modelling Relation It is important to recognise that models are not unique descriptions of a particular system, there is no such thing as a unique and complete model. The idea that such a perfect theory describing the system exactly exists is a widespread belief. The idea is based on the

19 2.4 CHAPTER 2. MODELLING OF DYNAMIC SYSTEMS IN THE TIME DOMAIN assumption that with growing sophistication the model eventually converges to the system. However the wrong conclusion is already indicated in this sentence, because the model does not converge to the system itself, but in the best case its ability to describe the behaviour of the modelled system increases until no difference between the two behaviours can be observed within the limits of the experimentally accessible region. Forming a set of models with increasing sophistication, the cost function decreases measuring the ability of the model to describe the behaviour of the system. A very illustrative example for such a set of models is a set of physical pictures of an object with increasing sophistication : caricature, drawing, painted picture, black and white photography, colour photography, hologram, sculpture... Models do not come in isolation, but once a model has been isolated it can be studied in its own right and for the sake of its own properties. In general there is this hierarchical set of models with increasing degree of sophistication each one adapted to a particular set of conditions and a purpose. The conditions under which the system is operating and for which the model shall be valid will have an impact on the degree of sophistication. The purpose also makes its contribution to the choice of the model because the model must only be as good as the particular application requires. Thus a comparably simple model in the form of algebraic equations might be sufficient for describing the steady state behaviour of a system, but a set of ordinary differential equations may be required for control. On the other hand the same purpose, say control, might require a higher order differential equation under certain conditions but a lower order model would suffice under others. (Aris 1978 [2], Aris 1979 [3]). Apostel 1960 [1] comprised these considerations in a relation describing modelling by defining = R(S, P, M, T ) (2.1) The subject S takes, in view of the purpose P,theentityM as a model for the prototype T. Model and prototype can belong to the same class of entities or to different classes of entities. They can for example be both images or both perceptions or drawings, formalisms (calculi), languages or physical systems. However, they can also belong to two different entities so can M be a image and T a physical system, or inversely, M can be a language and T acalculus, or inversely etc. Apostel s attempt at formalising the modelling relation in its philosophical aspects is not too much of interest to the engineering community, but every scientist or engineer should bear in mind that, when ever he decides to model a system, that he is modelling the system given certain conditions and that the model will be used in a particular application. Both influence the choice of the model and therefore the choice of the theory. The model is the representation of the essential aspects of the system (where essential is a subjective term), which presents the knowledge about the system in usable form. So there is no value in a model that is too complicated to be used. Aris 1978 [2], Aris 1979 [3] gives a graphical summary of a set of models describing the same system, where the sophistication of the model is a function of the conditions and the purpose, which is in essence Apostel s modelling relation.

20 2.2. MODELLING: A DESIGN ACTIVITY The Concept State Mathematical system theory is very closely related to the term state space theory, but even though the meaning of the term state is intuitively clear, a precise definition is quite difficult in particular because the term is used in different disciplines of natural sciences : Kalman 1963 [15], one of the leading people in mathematical system theory, wrote in 1963: The state is to be regarded always as an abstract quantity. Intuitively speaking, the state is the minimal amount of information about the past history of the system which suffices to predict the effect of the past upon the future. Kailath 1980 [14]: The state provides a sufficient statistic so to say, that enables us to calculate the future response to a new input without worrying about previous inputs. Note also that more than one past input can lead to the same state. Therefore, the state is really a minimal sufficient statistic. It contains just enough information, no less and no more, to enable us to calculate the future responses without further reference to the old history of inputs and responses as in more colloquial usage, the knowledge of the state vector at any time specifies the state or condition of the system at that time. Kailath also discusses a mathematical derivation which is due to Nerode 1958 [18] specifying the meaning of the term state more precisely. As a chemical engineer talking about states, it is interesting to compare our common understanding of the term state, as it is used in thermodynamics, with the usage in mathematical system theory : Denbigh 1971 [6] uses it as a primitive without really explaining its meaning. Moulines (1978) cites Falk & Jung (1959) which translated reads : In an axiomatic description the state plays the rôle of a not precisely defined basic object of the theory. The only condition is that the states are distinguishable objects. The latter definition is extremely abstract and reflects a little bit of the difficulty that people have to define terms precisely capturing all aspects of their common usage. But, what are the conclusions? There are in principle three aspects of the term state that one should keep in mind : 1. states are distinguishable objects - no two states are the same 2. states are independent of the history - exact differentials 3. states contain all the information about the condition (state) of the system at a given time. 2.2 Modelling: A Design Activity Mathematical models are used widely in engineering and science. Models are an abstraction of the plant and can thus be used as a substitute for the plant in any study aiming at abstract results associated with the plant. This includes activities associated with design and operation of a plant. Needless to say that modelling takes a central role in any engineering and science curriculum and is an important activity in industrial research and development. The subject of these lectures is mechanistic modelling of plants and processes to the extent possible. Models will thus build on first principles, namely the conservation of extensive quantities. Thus the system s concept is underlying all the derivations, systems being either of the class physical or abstract. Modelling is a design activity. For any given plant an infinite number of models can be

21 2.6 CHAPTER 2. MODELLING OF DYNAMIC SYSTEMS IN THE TIME DOMAIN generated none of which will exhibit the same behaviour as the plant. Any model must focus on a specific aspect of the plant that is of interest to the application of the model. This implies that the model is directly associated with the application. We shall not expand on this subject any further, though an excursion into philosophy of science would be interesting indeed. For now it is only essential to establish that modelling is a design activity, which requires human input. Also, the fact that we deal with a design activity and not with a purely analytical mapping makes the discussion somewhat ambiguous. The text tries to limit ambiguity to the extent possible and where it is not possible, it is hoped that it will become sufficiently clear to the reader that indeed one enters somewhat unsafe grounds. Modelling as an application-oriented research subject has been with this group for nearly two decades. The main results are probably in the level of abstraction that has been achieved as it allows other people to accelerate their learning curve as they enter the activity. The text is certainly not complete, in fact it is a patchwork of pieces taken from various sources. The integration has not been completed but it is hoped that the reader will find the material instructive What is Modelling? Different people employ a different definition of modelling depending on their particular view and use. A (mathematical) model may be any of the ones below: 1. Primary mathematical model: The model is given as a set of differential equations and a set of algebraic equations in symbolic form. 2. Derived mathematical models: Linearisation, model reduction and a set of other assumptions such as steady-state assumptions result in modified models. 3. Instantiated mathematical model: The mathematical problem has been specified by instantiating a set of variables and signals. 4. Coded model: The instantiated model has been translated into a computer language and pieces of code were added providing the interface and thus allowing the code to be embedded in the code of the problem solver. 5. Integrated (solved) model: The results from integrating (solving) the model for the unknown quantities. This course focuses on the first, namely the design and construction of the primary model. Before we move to this subject, though, a few words regarding the other interpretations or, as they are presented here, stages of modelling. The ability of manipulating the model casting it into a slightly different form through the application of assumptions allows for a multi-facetted modelling approach. Some of these assumptions will be discussed to some detail the corresponding chapter. The instantiated model is the result of going through the problem specification stage. The objective is to define, and later solve, a particular problem (design problem, simulation problem, etc.) by filling in the known things. The objective of this stage is to generate a well-posed mathematical problem for which a solution exists. This problem has multiple facettes. Ultimately, though, one would like to guarantee solvability of the posed problem. Unfortunately it is almost always nontrivial or impossible to prove solvability of a set of DAE models at this early state. Indeed this subject is a source for interesting research projects. The problem of translating and splicing the instantiated models into a piece of usable code is a pure computer science task. In most cases this task is rather straightforward given a

22 2.3. PRIMITIVE MODEL COMPONENTS 2.7 well-educated person is available for the job. As mentioned before, we shall limit our discussion to the first stage, namely establishing a primary model and to some extent to the domain of assumptions, which also brings in aspects of the second stage A Step-Wise Approach to Primary Modelling Modelling, as we apply it to physical-chemical and biological processes, splits in a number of steps some of which we will discuss in more detail some of which we will pass over, as they are rather straightforward. The steps are not uniquely defined, but a useful set is 1. Structure process: A recursive process of defining elements representing the plant s behaviour. 2. Identify exchanges of extensive quantities between these process components 3. Define chemistry of the process. 4. Define details such as transfer laws, kinetics, geometrical relations, relations between extensive and intensive quantities (definitions of intensive quantities) and relations between intensive quantities (equations of state) 5. Define control loops 6. Assemble model The result of step 1 and 3 is a topology that describes the containment of the plant with its capacities and their interactions. Step 3 overlays the species topologies, thus what species can be expected where in the process. The subject of topologies will be discussed in Chapter 3. Chapter 2 describes the basics of the primitive system components and their characteristics with a focus on describing their behaviour in terms of mass and energy but not momentum and other extensive quantities that may be of interest. Most of what is in step 3 is common knowledge and is viewed here to be present as prerequisites. Some material on the production term has been introduced, as it is specific to the chemical nature of the process and thus cannot be expected to be known by all participants. The subject of control loops will not be discussed here and the last step is obviously trivial in terms of the activity though it is quite a data-handling job if one deals with large numbers of systems components. 2.3 Primitive Model Components The approach taken here is based on physical insight and aims at state space models. Statespace models are the result of applying the systems concept to modelling natural processes using basic concepts of macroscopic thermodynamics. Thus a macroscopic view is taken, neglecting the quantised nature of mass and energy. As a result, capacities are thermodynamic systems in the classical sense and field theory applies for the transfer between capacities. The definition of a physical system splits the universe into two parts, namely the system and its environment. Together they make up all that exists, namely the universe. The system is separated from its environment by its boundary and a system is recursively subdivided into subsystems, thus forming a tree of systems, each of which is separated from its environment by its boundary. The definition of the boundary and system is thus circular. Systems exchange extensive quantities through their common boundary. Figure 2.1 shows a crude abstraction of a universe in pictorial form.

23 2.8 CHAPTER 2. MODELLING OF DYNAMIC SYSTEMS IN THE TIME DOMAIN Environment Universe system boundary system (plant) Figure 2.1: Subdivision of a Universe into a system and its environment A model of a system is the assembly of models of its primitive components, which is the subject of this chapter Basic Definitions As long as one stays on the macroscopic scale, that is, more than roughly 1000 basic elements are included in the characteristic length, the behaviour of processes may be represented by two principle components. The first represents the primitive capacitive elements, the simple systems and second component describes the communication between systems, represented by connections. Subdividing the spatial domain occupied by the plant into junks is mapped into showing the process as a network of capacities represented by simple thermodynamic systems and connections. The state-space model for the overall system is then the agglomeration of the state-space models for the simple systems. Thus state-space models for the simple systems must be established. State-space models for single-phase systems are available as they are described by the concepts of thermodynamics-in essence the conservation principles applied to extensive quantities relevant for the description of the system. We thus choose the primitive system as a single-phase system: Definition - Simple Physical System Σ s : Body of finite or differential volume consisting of a single phase or pseudo-phase. The refinement simple relates to the predicate single uniform phase or pseudo phase. If the volume is of finite dimensions with spatially uniform intensive properties, the system is called a lumped system. Mathematically, the model is a set of ordinary differential equations. If the intensive properties are not uniform over the extent of the phase, a infinitely small volume element must be chosen which results in a mathematical description of the form of partial differential equations including not only variations in time but also in space. This second class of primitive systems is also called distributed systems. In essence, the concept physical system imposes the condition of a capacity of any nature to be present and in addition the

24 2.3. PRIMITIVE MODEL COMPONENTS 2.9 condition that the state of the system is uniform within the spatial domain it occupies 1. Definition - Phase ψ : Spatial domain with uniform physical properties. The extension to pseudo phase allows for averaging operations resulting in pseudo phases with averaged intensive properties. This process allows making adjustments in the desired granularity of the representation. On a fine scale, the phase concept applies usually well. As one increases the granularity, that is, larger spatial domains are represented as simple systems, the system may include matter in different state of aggregation or different materials. Nevertheless, for the purpose of the model, the whole may be viewed as a uniform system. For this purpose the concept of pseudo phase is introduced: Definition - Pseudo Phase ψ : Spatial domain with spatially averaged properties. The definition of system and its boundary cannot be separated. They are circularly coupled. The boundary of the system is a theoretical artefact. It has no capacity. Because boundaries often replace physical walls, people often interpret the boundary as a capacitive component of the system. This is not the case. Mapping a physical wall into a boundary of a thermodynamic system implies that the capacity effects of the wall are neglected and that it is mapped into an abstract thermodynamic wall. Definition - Boundary Ω : Imaginary wall of no mass separating adjacent systems. A connection represents a communication path coupling the capacitive elements, the systems. The generic connection existing between the systems Σ a and Σ b through the interface element Ω i is denoted by c a i b. Definition - Connection c a i b : Directed communication path between system Σ a and system Σ b through common boundary element Ω i. c a 1 b Ω 2 Σ a Ω 1 c b 2 a Σ b Σ a c a 1 b c b 2 a Σ b Figure 2.2: Representation of processes as network of primitive systems and connections Figure 2.2 shows a process consisting of two communicating systems Σ a and Σ b. The common boundary is split into two elements. The two systems communicate trough each of the two 1 Webster defines phase as follows: Phase n [NL phasis, fr. Gk, appearance of a star, phase of the moon]:... 4: a homogeneous, physically distinct, and mechanically separable portion of matter resent in a nonhomogeneous physicochemical system;... The reader may note that the here underlying phase concept implies mechanical separation of phases.