Feedback systems

Often we are interested in finding and characterizing the evolution of patterns over time. This course introduces formal tools to design, model, and analyze evolving interconnected systems in which information, data sensing, and decision-making mechanisms strongly couple two or more subsystems. Formal techniques include frequency-domain and time-domain methods.

The alternative to deterministic predictability,  when it comes unachievable,  comes from feedback…Jeff G. Bohn


Instructors: Jorge Finke / Juan Manuel Nogales

Level: undergraduate


Date (week starting on…)Recitation (Mon)Lecture (Wed.)Labs (Fri)   
July 26 (week 1)Lec 1.1 (intro)Lec 1.2 (modeling)
Aug. 2 (week 2)Lec 2.1 (equilibria)Lec 2.2 (stability)
Aug. 9 (week 3)Lec 3.1 (Lyapunov)Lec 3.2 (Lyapunov proof) - HW1 dueLab 1
Aug. 16 (week 4)-Lec 4.1 (Lyapunov functions) - HW2 due
Aug. 23 (week 5)ProjectLec 4.2 (Lasalle) - HW3 dueLab 2
Aug. 30 (week 6)ProjectLec 5.1 (linearization) - HW4 due
Sept. 6 (week 7)Lec 5.2 (stability LS)Lec 6.1 (reachability) - HW5 dueLab 3
Sept. 13 (week 8)ProjectLec 6.2 (feedback controller)
Sept. 20 (week 9)Midterm ILec 7.1 (observability) - HW6 due
Sept. 27 (week 10)ProjectLec 7.2 (the observer) - HW7 due
Oct. 4 (week 11)Lec 8.1 (TF func.)Lec 9.1 (Bode) - HW8 dueLab 4
Oct. 11 (week 12)ProjectLec 10.1 (Nyquist) - HW9 due
Oct. 18 (week 13)-
Lec 11.1 (PID)
Oct. 25 (week 14)ProjectLec 12.1 (PID)Lab 5
Nov. 1 (week 15)-Lec 13.1 (PID) HW10 due
Nov. 8 (week 16)Final reviewQ&A

 If you taking this course for credit, please fill out this form. After registering,  login and click on “enroll for this course.” You should  now be able to submit homework solutions online. Quizzes will be based on homework problems.


To access the problem sets and lab assignments please use the password provided in class. 


Homework sets

HW1 - Introduction to feedback  (submit)

HW2 - Modeling dynamical systems  (submit)

HW3 - State-space representations, equilibria and stability (submit)

HW4 - Lyapunov analysis (submit)

HW5 - Krasovskii-Lasalle’s Theorem  (submit)

HW6 - Linearization (submit)

HW7 - Reachability (submit)

HW8 - Observability  (submit)

HW9 - Transfer function, block algebra, root locus and Hurwitz stability criterion (submit)

HW10 - Nyquist and PID controller (submit)


Dynamics and modeling

What a is a dynamic model? What does a model say about a system? Define concepts of state, dynamics, inputs and outputs. Non-linear vs. linear systems; overview modeling techniques.

Lasalle theorem

Krasovskii-Lassale Invariance Principle; Lyapunov functions for linear systems.


Compute linearization of a nonlinear system around an equilibrium point; Lyapunov Indirect Method; Examples.

State feedback

Define reachability of a system; test for reachability of linear systems; state feedback for linear systems

Output feedback

Define observability; conditions for linear systems; state estimation; examples.

Transfer functions

Content: Transfer function; Routh-Hurwitz criterion; Canonical form realizations

PID (part 1)


PID (proportional–integral–derivative controller); basic properties; PID implementation.

Course review


Validation of linear time-invariant models using barrier certificates