Lecture 6 - Fitting models to data

In Lecture 4 we reviewed how to extract the Gain, Time Constant, and Dead-time from a step test. There are many situations when it is not possible or doesn't make sense to generate a complete step test.

A FOPDT model can be obtained by fitting the model to data. This allows other forms of step testing such as:

  • Pulse: step up followed by step back to original value
  • Doublet: two pulses in opposite directions
  • Pseudo-Random Binary Sequence (PRBS): steps of varying frequency and magnitude
  • Smart-stepping: optimize steps to extract the most information while keeping the process within bounds

In practice, smart-stepping or PRBS signals are used for identification of models for multivariable control. For tuning of PID controllers a step, doublet, or pulse test is often preferred.

Linear vs. Nonlinear Models

At the end of class we ran through an example of a gravity drained tank.

Fig 1: Diagram of the Gravity Drained Tank

Fig 2: Sequence of Valve Movements to Test Models

Half of the class derived a FOPDT model of the process using empirical fitting techniques. The other half of the class used a material balance to obtain a model. A comparison of the two models is shown below:

Fig 3: Linear Model (FOPDT). The linear response is easy to fit to the data but deviates, especially during the periods that are far from the steady state values.

Fig 4: Nonlinear Model Based on a Material Balance. The nonlinear response is valid over a wider range of operation.


  1. Course reading for next class: Chapter 7-9 (PPC).
  2. Assignment due by the start of Lecture #7: SP3

You'll use your answers from Q3.1 and Q3.2 on SP3. I've included the answers to Q3.1 and Q3.2 for your reference.

Don't forget that the first part of each homework assignment is to write a sentence or two stating the purpose of the problem and what concept is reinforced.

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