Classical process control builds on linear ordinary differential equations (ODEs) and the technique of the Laplace transform. This is a topic that we no doubt have come across in an introductory course on differential equations – like two years ago? Yes, we easily have forgotten the details. Therefore an attempt is made here to refresh the material necessary to solve control problems; other details and steps will be skipped. We can always refer back to our old textbook if we want to answer long-forgotten but not urgent questions.

What Are We Up to?

• The properties of Laplace transform and the transforms of some common functions. We need them to construct a table for doing an inverse transform.

• Because we are doing an inverse transform by means of a look-up table, we need to break down any given transfer functions into smaller parts that match what the table has – what are called partial fractions. The time-domain function is the sum of the inverse transform of the individual terms, making use of the fact that Laplace transform is a linear operator.

• The time-response characteristics of a model can be inferred from the poles, i.e., the roots of the characteristic polynomial. This observation is independent of the input function and singularly the most important point that we must master before moving onto control analysis.

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