CONTENTS

Any description of a system could be considered to be a model of that system. Although the ability to encapsulate dynamic information is important, some analysis and design techniques require only steady-state information. Models allow the effects of time and space to be scaled, extraction of properties and hence simplification, to retain only those details relevant to the problem. The use of models therefore reduces the need for real experimentation and facilitates the achievement of many different purposes at reduced cost, risk and time.

In terms of control requirements, the model must contain information that enable prediction of the consequences of changing process operating conditions. Within this context, a model could either be a mathematical or statistical description of specific aspects of the process. It can also be in the form of qualitative descriptions of process behaviour. A non-exhaustive categorisation of model forms is shown in Fig. 1. Depending on the task, different model types will be employed.

Figure 1. Classification of Model Types for Process Monitoring and Control

2.1. Mechanistic Models

If much is known about the process and its characteristics are well defined, then a set of differential equations can be used to describe its dynamic behaviour. This is known as 'mechanistic' model development. The mechanistic model is usually derived from the physics and chemistry governing the process. Depending on the system, the structure of the final model may either be a lumped parameter or a distributed parameter representation. Lumped parameter models are described by ordinary differential equations (ODEs) while distributed parameter systems representations require the use of partial differential equations (PDEs). ODEs are used to describe behaviour in one dimension, normally time, e.g. the level of liquid in a tank. PDE models arise due to dependence also on spatial locations, e.g. the temperature profile of liquid in a tank that is not well mixed.

Obviously, a distributed parameter model is more complex and hence harder to develop. More importantly, the solution of PDEs is also less straightforward. Nevertheless, a distributed model can be approximated by a series of ODEs given simplifying assumptions. Both lumped and distributed parameter models can be further classified into linear or nonlinear descriptions. Usually nonlinear, the differential equations are often linearised to enable tractable analysis.

In many cases, typically due to financial and time constraints, mechanistic model development may not be practically feasible. This is particularly true when knowledge about the process is initially vague or if the process is so complex that the resulting equations cannot be solved. Under such circumstances, empirical or 'black-box' models may be built using data collected from the plant.

2.2. Black Box Models

Black box models simply describe the functional relationships between system inputs and system outputs. They are, by implication, lumped parameter models. The parameters of these functions do not have any physical significance in terms of equivalence to process parameters such as heat or mass transfer coefficients, reaction kinetics, etc. This is the disadvantage of black box models compared to mechanistic models. However, if the aim is to merely represent faithfully some trends in process behaviour, then the black box modelling approach is just as effective. Moreover, the cost of modelling is orders of magnitude smaller than that associated with the development of mechanistic models.

As shown in Fig. 1, black box models can be further classified into linear and nonlinear forms. In the linear category, transfer function and time series models predominate. With sampled data systems, this delineation is, in a sense, arbitrary. The only distinguishing factor is that in time-series models, variables are treated as random variables. In the absence of random effects, the transfer function and time-series models are equivalent. Given the relevant data, a variety of techniques may be used to identify the parameters of linear black box models [Eykhoff, 1974]. The most common techniques used, though, are least-squares based algorithms.

Under the nonlinear category, time-series feature again together with neural network based models. In nonlinear time-series, the nonlinear behaviour of the process is modelled by combinations of weighted cross-products and powers of the variables used in the representation. The parameters of the functions are still linear and thus facilitates identification using least squares based techniques. Neural networks are not new paradigms to nonlinear systems modelling. However, the increase in cheap computing power and certain powerful theoretical results have led to a resurgence in the use of neural networks in model building [Cybenko, 1989; Lippmann, 1987, Rummelhart and McCelland, 1986].

2.3. Qualitative Models

There are instances where the nature of the process may preclude mathematical description, e.g. when the process is operated at distinct operating regions or when physical limits exist. This results in discontinuities that are not amenable to mathematical descriptions. In this case, qualitative models can be formulated. The simplest form of a qualitative model is the 'rule-based' model that makes use of 'IF-THEN-ELSE' constructs to describe process behaviour. These rules are elicited from human experts. Alternatively, Genetic Algorithms and Rule Induction techniques can be applied to process data to generate these describing rules [South et al, 1993]. More sophisticated approaches make use of Qualitative Physics theory [Bobrow, 1984; Weld and deKleer, 1990] and its variants. These latter methods aim to rectify the disadvantages of purely rule based models by invoking some form of algebra so that the preciseness of mathematical modelling approaches could be achieved.

Of these, Qualitative Transfer Functions (QTFs) [Feray Beaumont et al, 1992] appear to be the most suitable for process monitoring and control applications. QTFs retain many of the qualities of quantitative transfer functions that describe the relationship between an input and an output variable, particularly the ability to embody temporal aspects of process behaviour. The technique was conceived for applications in the process control domain. Cast within an object framework, a model is built up of smaller sub-systems and connected together as in a directed graph. Each node in the graph represents a variable while the arcs that connect the nodes describe the influence or relationship between the nodes. Overall system behaviour is derived by traversing the graph, from input sources to output sinks.

Models derived based on the use of Fuzzy Set theory can also be classified as qualitative models. Proposed by Zadeh [1965, 1971], fuzzy set theory contains an algebra and a set of linguistics that facilitates descriptions of complex and ill-defined systems. Magnitudes of changes are quantised as 'negative medium', 'positive large' and so on. The model combines elements of the rule based and probabilistic approaches and sets of symbols with interpretations such as, 'If the increment of the input is positive large, the possibility of the increment on the output being negative small is 0.8'. Fuzzy models are being used in everyday life without our being aware of their presence, e.g. washing machines, auto focus cameras, etc.

2.4. Statistical Models

Describing processes in statistical terms is another modelling technique. Time-series analysis which has a heavy statistical bias may be considered to fall into this model category. Nevertheless, due to its widespread and interchangeable use in the development of deterministic as well as stochastic digital control algorithms, the earlier classification is more appropriate. The statistical approach is made necessary by the uncertainties surrounding some process systems. This technique has roots in statistical data analysis, information theory, games theory and the theory of decision systems.

Probabilistic models are characterised by the probability density functions of the variables. The most common is the normal distribution which provides information about the likelihood of a variable taking on certain values. Multivariate probability density functions can also be formulated but interpretation becomes difficult when more than two variables are considered. Correlation models arise by quantifying the degree of similarity between two variables by monitoring their variations. This is again quite a commonly used technique, and is implicit when associations between variables are analysed using regression techniques.

System dynamics are not captured by statistical models. However, in modern control practice, they play an important role particularly in assisting in higher level decision making, process monitoring, data analysis and obviously, in Statistical Process Control.