{"id":2157,"date":"2023-12-05T13:49:27","date_gmt":"2023-12-05T13:49:27","guid":{"rendered":"https:\/\/blogs.zeiss.com\/digital-innovation\/en\/?p=2157"},"modified":"2023-12-05T13:50:52","modified_gmt":"2023-12-05T13:50:52","slug":"data-driven-process-control-part-2","status":"publish","type":"post","link":"https:\/\/blogs.zeiss.com\/digital-innovation\/en\/data-driven-process-control-part-2\/","title":{"rendered":"Data-driven Process Control – Part 2: Modelling System Behaviour"},"content":{"rendered":"\n

After the first article in this series<\/a> dealt with the general basic building blocks of control systems, the second article is dedicated to modeling the behavior of systems. The focus here is on differentiating between different types of modeling. The main part of the article introduces a special data-driven approach that has recently attracted growing scientific interest.<\/p>\n\n\n\n\n\n

Knowledge of the system behaviour, i.e. knowledge of the quantitative change in output when the system\u2019s inputs change, is a basic prerequisite for system control. This knowledge is represented by behavioural models, the development of which we will examine in more detail in this section.<\/p>\n\n\n\n

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A physical example system<\/h2>\n\n\n\n

As a simple physical example, let\u2019s look at what is known as an ideal planar pendulum. The pendulum mass is assumed to be point-like and suspended on a massless cord; all frictional forces are disregarded. The only force acting on the mass is the gravitational force of the Earth (see Figure 3).<\/p>\n\n\n\n

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Figure 3: The ideal pendulum<\/em><\/figcaption><\/figure>\n\n\n\n
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We will use this example to describe three basic types of modelling.<\/p>\n\n\n\n

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Whitebox Modelling<\/h2>\n\n\n\n

In this type of modelling, the system behaviour is represented by differential equations whose parameters are completely known. The approach is highly application-specific and requires a high degree of detailed knowledge of the system under consideration. The complexity of real-life systems obviously places limits on the application of this approach. In addition, the strategy is difficult to automate and the models ob<\/p>\n\n\n\n

We will use this example to describe three basic types of modelling.tained are difficult to adapt to changing requirements.<\/p>\n\n\n\n

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Example \u2013 white box modelling of the pendulum system<\/p>\n\n\n\n

The starting point is Newton\u2019s second law \"F = m \, a\" and Newton\u2019s law of gravitation, specifically for bodies near the Earth\u2019s surface \"F = m \, g \, [\, 0, -1 \,]^T\". In terms of the motion of the pendulum, only the force \"F_T\" acting tangentially on the pendulum mass is to be considered, since the radial component is compensated by the cord. For the same reason, only the tangential acceleration \"a_T\" is relevant. Using the time-dependent deflection angle \"\theta(t)\" and the angular acceleration \"\ddot{\theta}(t)\" (see Figure 3), the variables \"F_T\" and \"a_T\" are given as   <\/p>\n\n\n\n

  <\/span>   <\/span>\"\[F_T = -m \ g \sin \theta(t) \quad \text{and} \quad a_T(t) = l \, \ddot{\theta}(t)\]\"<\/p><\/p>\n\n\n\n

with the length \"l\" of the pendulum cord and \"g \approx 9.81 \frac{m}{s^2}\" as the gravitational acceleration on Earth. Therefore, \"m \ a_T = F_T\" results in the following differential equation for the deflection angle \"\theta\"<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[\ddot{\theta} (t) = - \frac{g}{l} \ \sin \theta(t)\]\"<\/p><\/p>\n\n\n\n

If we additionally assume that the angle \"\theta\" is small, we obtain as a further simplification

  <\/span>   <\/span>\"\[\ddot{\theta} (t) = - \frac{g}{l} \ \theta(t)\]\"<\/p><\/p>\n\n\n\n

This differential equation can be used to predict the behaviour of the pendulum for any initial angle \"\theta_0 \ll 1\" and any initial velocity \"\dot{\theta}_0\".<\/p>\n\n\n\n

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Grey- and Blackbox Modelling<\/h2>\n\n\n\n

This modelling approach also results in differential or difference equations, which, however, only specify the basic structure of the system and therefore contain free parameters. If the system model incorporates prior knowledge, for example in the form of physical principles, then it is a grey box model. Otherwise it is referred to as a black box model.<\/p>\n\n\n\n

The free parameters of the model are determined from observed data of the system during an adaptation step, whereby this step of the modelling can be automated to a large extent. The choice of model structure, however, is critical for the quality of the model, requires a high degree of experience due to its complexity and can therefore only be automated to a limited degree. Moreover, in contrast to the white box approach, grey and black box modelling can be applied to systems of any complexity.<\/p>\n\n\n\n

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Data-Based Modelling<\/h2>\n\n\n\n

This new approach has been attracting an increasing amount of attention for some years now. The starting point is the notion of identifying a system solely on the basis of its externally verifiable behaviour. A detailed description of the systems-theoretical background for this approach can be found in [1]<\/a>. With the increasing availability of process data from technical systems, the view of this data-driven approach has changed. It was recognized that the description of system behavior based purely on observations opened the door to new models and algorithms (see [2]<\/a> and [3]<\/a>). This transition is comparable to developments in the application of neural networks in recent years.<\/p>\n\n\n\n

Since it is an unsupervised learning procedure for non-parametric system representations, no model of the system is constructed, unlike in white, grey and black box modelling. As such, the applicability of this approach is not subject to complexity-related limitations, nor does the need for a structural decision prevent it from being automated. That said, the approach is initially limited to so-called linear and time-invariant systems (see \u201cMathematical background\u201d).<\/p>\n\n\n\n

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Example \u2013 Data-based modelling of the pendulum system<\/p>\n\n\n\n

To arrive at a data-based representation for the pendulum system, we only need to conduct two experiments. Using the notation \"x(t) = [\, \theta(t), \dot{\theta}(t) \, ]^T\" with the deflection angle \"\theta(t)\" and the angular velocity \"\dot{\theta}(t)\", the initial conditions for the two experiments can be taken as<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[x_1(0) = \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} \quad \text{and} \quad x_2(0) = \begin{bmatrix} 0 \\ 1 \\ \end{bmatrix}\]\"<\/p><\/p>\n\n\n\n

The first experiment records the motion \"x_1(t)\" of the pendulum at the starting angle \"1^{\circ}\" and vanishing starting velocity at discrete points in time \"\{ t_i \}_{i=1}^n\". The second experiment uses the inverse setting for recording \"x_2(t)\".<\/p>\n\n\n\n

The recordings of both experiments can then be used to calculate any other pendulum motion<\/span> \"x(t)\" <\/span>at the initial conditions <\/span>\"x(0) = [\, \theta_0, \dot{\theta}_0 \,]^T\" by means of the relationship<\/span><\/p>\n\n\n\n

  <\/span>   <\/span>\"\[x(t) = \theta_0 \ x_1(t) + \dot{\theta}_0 \ x_2(t)\]\"<\/p><\/p>\n\n\n\n

at the points in time \"\{ t_i \}_{i=1}^n\", whereby the linear independence of the vectors of the two initial conditions is essential. Further we obtain the compact representation<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[B \, x(0) = x \quad \text{with} \quad B = \begin{bmatrix} x_1(t_1) & x_2(t_1) \\ \vdots & \vdots \\ x_1(t_n) & x_2(t_n) \end{bmatrix} \quad \text{and} \quad x = \begin{bmatrix} x(t_1) \\ \vdots \\ x(t_n) \end{bmatrix}\]\"<\/p><\/p>\n\n\n\n

We call the matrix \"B\" an algebraic representation of the pendulum behavior.<\/p>\n\n\n\n

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The objective of reconstructing the entire system behaviour from observed data alone raises three main questions:<\/p>\n\n\n\n