Lecture 15 - Linearization

Now that we've derived nonlinear models based on material and energy balances, we need to get them into a form for linear systems analysis. We do this through a process called linearization. In Lecture 15, we review the mathematics of how to linearize a nonlinear function. We have a couple exercises to help practice the linearization process.

• Linearization Notes
• Worksheet on Linearization
• Worksheet on Linearization Solution

Linear vs. Nonlinear Models: CSTR Case Study

• CSTR Nonlinear vs. Linear MATLAB Solution

Linear vs. Nonlinear Models: Gravity Drained Tank Case Study

We previously 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

We derived a FOPDT model of the process using empirical fitting techniques. A first principles approach was also used to obtain a model from a material balance. 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.

Discussion Questions

Test the lecture's ideas before or after class - commit to an answer first, then check it with a classmate or an AI tutor.

1. A student linearizes the CSTR model around a point that is not a steady state of the nonlinear equations. What specific term fails to vanish, and what nonsense does the resulting "linear model" predict at zero input?
2. The linearized gravity-drained tank model and the nonlinear model agree for a 2% valve step but diverge badly for a 30% step. Explain, using the Taylor series, exactly what was thrown away and why the error grows with step size.
3. Why do the deviation variables matter? What goes wrong if you drop the primes and treat the linear model as relating absolute temperature to absolute flow?
4. Control design almost always starts from a linear model even though every real process is nonlinear. Give two practical reasons this is defensible, and one situation from the CSTR case study where it is not.

App: The Stirred Reactor Control Studio runs the full nonlinear exothermic CSTR - move the operating point and watch the effective process gain change, which is exactly the behavior a single linearization cannot capture.