# AME 30315 – Differential Equations, Vibrations, and Controls II

## Overview

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.

### Topics

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)

### Text:

Engineering Differential Equations: Theory and Applications, Goodwine, 2011

### Lectures:

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
23. Lag compensator design
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)