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## 2. Introduction to the Principles of Feedback

#### 2.3.2 Modeling

To make progress on the control-system design problem as set out above, it is first necessary to gain an understanding of how the process operates. This understanding is typically expressed in the form of a mathematical model that describes the steady-state and the dynamic behavior of the process. To construct such a model, we first define relevant process variables. Thus, we introduce the following:

 : commanded level of steel in mould : actual level of steel in mould : valve position : casting speed : inflow of matter into the mould : outflow of matter from the mould.

Physics suggests that the mould level will be proportional to the integral of the difference between in- and outflow:

 (2.3.1)

where we have assumed a unit cross-section of the mould for simplicity. We also assume, again for simplicity, that the measurements of valve position, v(t) and casting speed, , are calibrated such that they actually indicate the corresponding in- and outflows:

 (2.3.2)

 (2.3.3)

Hence, the process model becomes

 (2.3.4)

The casting speed can be measured fairly accurately, but mould-level sensors are typically prone to high-frequency measurement noise, which we take into account by introducing an additive spurious signal n(t):

 (2.3.5)

where hm(t) is the measurement of h(t) corrupted by noise. A block diagram of the overall process model and the measurements is shown in Figure 2.3.

This is a very simple model, but it captures the essence of the problem.

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