DYNAMIC MODELLING APPROACHES

The availability of a rigorous fundamental dynamic model, as in chapter 5, is rare. Although a phenomenological structure exists for parameter estimation via data reconciliation methods, an inappropriate model structure may make such a fit impractical.

In industrial practice it is seldom technically or economically feasible to construct detailed first principles models. The industrial success of MPC is largely attributed to the availability of commercial software for developing linear dynamic models from plant tests. No established method exists for identifying empirical non-linear models, due to the complexity of non-linear systems. Consequently, non-linear models are frequently based on fundamental models, derived from energy and material conservation laws (Henson, 1998; Morari and Lee, 1999).

In system identification, the prime concern is to find a suitable model structure wherein a good model fit may be obtained. Prior knowledge of physical relationships should be used in the model structure. Identifying concrete model relationships makes the model development task complex. Wherever possible, the problem should be reduced to parameter estimation, thereby making the system identification task less rigorous. Three model types exist (1) white-box models, (2) grey-box models and (3) black-box models. White-box models include fundamental models, where knowledge and physical insight describes the process behaviour perfectly. Grey-box models are sub-divided into (a) physical modelling and (b) semi-physical modelling. Grey-box physical modelling pertains to cases where a model structure is known, but certain parameters need to be estimated from experimental data. Grey-box semi-physical modelling suggests that certain measured input signals are used for constructing unstructured models such as neural networks. Black-box modelling implies that no prior physical insight is available and is typically grounded in unstructured regression models (Sjöberg et al., 1995).

6.1.1 Fundamental Models

Fundamental dynamic models derive from transient material, energy and momentum balances that are relevant to the process. Without spatial variations, the general form of non-linear phenomenological models is:

( )

( )

( )

where the ordinary differential equations (f), the algebraic equations (g) and the measurement equations (h) are functions of the n-dimensional state variable vector x and the m-dimensional manipulated variable vector u . y is the a p-dimensional

observability vector dictated by the available sensor measurements. Fundamental models have several advantages over non-linear empirical models. The constrained structure and limited parameters require less process data for development. Particularly, model parameters may be estimated from laboratory experiments as opposed to time-consuming plant experiments needed for empirical modelling. Also, fundamental models extrapolate well to operating regions not represented in the data set used for model parameterisation, provided the underlying assumptions remain valid. Effective extrapolation is important where processes have a wide operating region. Commercial dynamic simulators, such as AspenPlus (from Aspen Technologies) and Hysys (Hyprotech), provide for rapid development of rigorous non-linear dynamic models. However, these fundamental models must be validated with plant or laboratory data that reflect typical operating conditions. On-line validation involves placing the model in predictive mode in parallel with the process, whereupon large deviations between plant measurements and the model predictions may necessitate further modelling effort (Henson, 1998).

6.1.2 Empirical models

Numerous processes elude fundamental modelling, owing to a lack of process knowledge and suitable fundamental equation frameworks. Although fundamental modelling is conceivable for a large number of chemical and mineral processing operations, mechanistic model-based approaches may prove unreliable in their predictions. Many micro-phenomena that occur in complex processes (e.g., flotation) are poorly understood and identifying the large number of parameters within a reasonable fundamental framework may require a vast number of dedicated experiments (Amirthalingam & Lee, 1997). Empirical non-linear models must

consequently be developed from dynamic plant input-output data. Discrete-time non- linear models include: (1) Hammerstein and Weiner models, (2) Volterra models, (3) polynomial ARMAX models and (4) artificial neural network models. Such non- linear input-output models have the general form:

( )

[

(

)

(

)

(

)

(

) ( )

(

)]

y (6-4)

where F is a non-linear mapping, k the sample index, y the measured variable, u the manipulated variable and e the noise input. The number of past samples used is denoted by n. State space representations of the input-output data are also feasible. Typically multi-variable processes are modelled by a multi-input, single-output model for each measured variable (Henson & Seborg, 1995).

Non-linear identification entails the task listed in Table 6-1.

Table 6-1 - Non-linear system identification (Henson, 1998).

1. Input sequence design - selection of appropriate measured and manipulated variables that impact y(k).

2. Structure selection - selection of input parameters ny, nu and ne.

3. Noise modelling - estimation of the noise input e(k).

4. Parameter estimation - estimation of the model (i.e., mapping) parameters. 5. Model validation - prediction comparisons with plant data not used during

training.

Most of the tasks listed in Table 6-1 remain open-ended problems. The relative suitability of either Hammerstein, Weiner, ARMAX and neural network models for a given problem is not formally defined. An important issue is the design of plant tests that provide sufficient excitation for determining the dynamics without impacting production. The optimal number of delayed inputs ny, nu and ne are difficult to

determine, requiring complex techniques such as false nearest neighbours. A promising method for non-linear multi-variable identification is to determine state representations through appropriate projection (i.e., linear or non-linear) of input- output time-series data. Also, most practical problems require multi-input multi- output dynamic models. Combining MISO process models into a multi-variable model may not be effective. Non-linear models will inherently contain modelling errors in their process approximation, which need to be quantified in the controller design and analysis to ensure robust control (Morari and Lee, 1999).

Empirical non-linear models offer advantages over fundamental models, in that empirical dynamic models do not require detailed process understanding. Artificial neural networks are the most popular framework (Henson, 1998).

6.1.3 Hybrid models

Hybrid non-linear models combine the fundamental and empirical approach, which allows exploitation of the advantages of both methods. Typically, hybrid models use empirical models to estimate unknown functions in fundamental models, such as reaction rates in a fundamental reactor model. Hybrid models constrain the underlying physics within a fundamental framework, while estimating complex sub-systems using empirical approaches (Henson, 1998).

6.1.4 Batch modelling

Though fundamental, empirical and hybrid models have found application in control applications for continuous processes, few applications of model-based control to batch processing exist (Morari & Lee, 1999). Control objectives in batch processes are most frequently posed as tracking problems for time-varying reference trajectories over a defined batch time. During a batch, the process variables traverse wide regions of the state space that are characterised by varying degrees of non-linearity. The batch trajectory may only be in a particular region of the state space for a limited time period, limiting the representative data that may be collected for modelling the regional process dynamics. For effective modelling, a representative data set larger than for continuous process modelling is typically required. Favourably, batch processing is repetitive and hence process information may be exploited in a framework that allows the use of past batch data along with real-time data in the control system. Previous batches may be incorporated through state estimation within the predictive control computation. Run-to-run learning is fundamental to batch optimisation and control (Morari & Lee, 1999).

Developing an accurate batch model is thus considerably more difficult than for continuous operation. Owing to the high probability of an inaccurate dynamic model, a model-based control system is likely to have significant tracking error. The modelling exercise needs to include additional identification rigour and the modelling technique must maximise process knowledge from limited process information.