Lecture 31 - Stability Analysis

Lecture 31 discusses techniques for determining system stability. Of particular interest is how the system stability changes with controller tuning. In this lecture, we review 3 techniques for stability analysis including:

• Routh Arrays
• Root Locus Plots
• Bode Diagrams

There are other techniques such as Direct Substitution and Nyquist Plot Analysis that are not covered in this lecture.

• Lecture 31 Notes
• Stability Analysis with MATLAB
• CSTR Example in MATLAB

Homework Problem 11.14

• PDC 11.14 Materials (zip file)

Other Material

• Lecture 31 Worksheet

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. "All coefficients of the characteristic equation are positive, so the loop is stable." Why is that conclusion premature for third order and above, and what does the Routh array add that the coefficient check cannot see?
2. A process is open-loop stable; a student concludes no controller can destabilize it. Using the characteristic equation 1 + G_cG_p = 0, explain where closed-loop instability comes from and why increasing K_c moves the closed-loop poles at all.
3. Routh analysis of the CSTR loop gives K_cmax. Would you run a plant at 0.9 K_cmax? Name two model uncertainties from this course that eat into that margin, and pick a defensible operating fraction.
4. Root locus, Routh arrays, and Bode plots answer the same stability question three ways. For (a) finding the ultimate gain quickly, (b) seeing how poles migrate as K_c changes, and (c) handling a time delay without approximation - which tool fits each job, and why does the delay single out one of them?

App: Push a real loop to its limit in the Stirred Reactor Control Studio - raise the controller gain on the exothermic CSTR until the response begins to sustain oscillations, and compare that observed ultimate gain with what your Routh or direct-substitution calculation predicts for the linearized model.