AME 30315 – Differential Equations, Vibrations, and Controls II


This material is from the Spring 2023 offering of AME 30315, Differential Equations, Vibrations, and Controls II, taught at the University of Notre Dame. This semester-long course covers fundamentals in systems theory, emphasizing frequency-domain methods.

Learning Objectives

The objective of the course is for students to be able to design feedback control systems for mechanical, electrical, and fluid systems described by linear ordinary differential equations (ODEs). Students will learn how to use Laplace transforms to solve ODEs, and will analyze the input/output properties of linear systems. This understanding will serve as the basis to design control systems that meet performance specifications (e.g., bandwidth, rise time, etc.) in the time and frequency domain. Students will also learn how to solve systems of ODEs and design controllers using state-space methods. Homeworks and examples will consider both by hand and computer-aided workflows.


Systems Analysis: Laplace transform solution methods to differential equations, transfer functions, frequency domain analysis (Bode plots, root locus), stability criteria
Control Synthesis: Fundamentals of feedback control, block diagrams, frequency domain control synthesis (lead/lag compensators), PID control
State Space Intro: Methods for solving coupled systems of linear ODEs (forced and unforced), multiple- degree of freedom vibrations, feedback design (LQR and pole placement)


Engineering Differential Equations: Theory and Applications, Goodwine, 2011


  1. Review of complex numbers
  2. Laplace transforms (intro)
  3. Properties of Laplace transforms
  4. Solving differential equations via Laplace (basic examples)
  5. Solving differential equations via Laplace (repeat roots, discontinuous forcing)
  6. Transfer functions
  7. Step response
  8. Second-order systems
  9. Second-order systems (cont.)
  10. Pole placement via feedback (transfer function edition)
  11. Dominant poles
  12. Frequency response and Bode diagrams
  13. Exam review
  14. [Exam]
  15. Bode plots
  16. Feedback and block diagrams
  17. Block diagram examples
  18. PD control
  19. Routh criteria
  20. PID control
  21. Gain and phase margin
  22. Lead compensator design
  23. Lag compensator design
  24. Lead/lag example
  25. Exam review
  26. [Exam]
  27. Systems of first-order ODEs
  28. Initial conditions and phase portrait
  29. Systems of first-order ODEs (handling complex eigenvalues)
  30. Examples
  31. Modal analysis
  32. Modal analysis (cont.)
  33. Systems of first-order ODEs  (Handling repeat eigenvalues)
  34. Force solutions via a change of variables
  35. Exam review
  36. [Exam]
  37. Forced solutions via Laplace, and transfer functions via state space
  38. Pole placement (state space edition)
  39. Linear Quadratic Regulator (LQR)
  40. Wrap Up