Linear Systems
University of Wisconsin–Madison
This is a graduate-level course on linear dynamical systems with an emphasis on state-space modeling in both discrete and continuous time. Topics covered include equilibrium points and linearization, natural and forced responses, canonical forms and transformations, controllability and observability, control-theoretic concepts such as pole placement, stabilization, dynamic compensation, and the separation principle. This course presents material that should be fundamental knowledge for students pursuing research in systems/control theory, signal processing, or mechanical/electrical/industrial engineering. The official prerequisite is MATH 340. Unofficially, you should be comfortable with linear algebra and MATLAB, and preferably have taken an introductory systems/controls course (e.g. ECE 330, 332 or 334)
Lectures: Tue/Thu, 9:30am–10:45am, Engineering Hall (EH), Room 3418.
Instructor: Laurent Lessard.
Submitting assignments: Gradescope.
Discussion forum: Piazza.
Part I: State-space fundamentals
Lecture notes | Dates | Reading from textbook | |
1. | Introduction and state-space models | Sep 7 | pages 1-14 |
2. | General models and linearization | Sep 12 | pages 15-24 |
3. | Solution of state equations | Sep 14 | pages 48-54 and 64-70 |
4. | The impulse response | Sep 19 | pages 61-64 |
5. | Diagonalization and modal form | Sep 21 | pages 72-92 and 438-442 |
6. | Controllability | Sep 26 | pages 129-133 and 443 |
7. | Controllability part 2 | Sep 28 | pages 108-122 and 133-138 |
8. | Observability | Oct 3 | pages 149-174 |
Part II: Analysis topics
Lecture notes | Dates | Reading from textbook | |
9. | Minimality of SISO systems | Oct 10 | pages 122-127, 168-170, and 186-191 |
10. | Kalman canonical form | Oct 12 | see here |
11. | Intro to stability | Oct 17 | pages 198-211 |
12. | Lyapunov theory | Oct 19 | pages 210-217 |
13. | Norms and BIBO stability | Oct 31 | pages 218 and 444-446 |
14. | System norms | Nov 2 | pages 219-224 |
Part III: Design and feedback
Lecture notes | Dates | Reading from textbook | |
15. | State feedback | Nov 9 | pages 234-236, 250-254, and 263-267 |
16. | Eigenvalue selection and placement | Nov 14 | pages 237-262 |
17. | Steady-state tracking | Nov 16 | pages 268-278 |
18. | Observers | Nov 21 | pages 300-316 |
19. | Separation property | Nov 28 | pages 323-331 |
20. | Linear Quadratic Regulator | Nov 30 | pages 390-397 |
21. | Kalman filter | Dec 5 |
Homework assignments
Assignment | Due date | Extra files | Solutions | |
1. | Homework 1: state-space models | Sep 20 | solutions | |
2. | Homework 2: state-space solutions | Oct 5 | solutions | |
3. | Homework 3: realization theory | Oct 20 | solutions | |
4. | Homework 4: stability | Nov 3 | solutions | |
5. | Homework 5: state feedback | Dec 1 | solutions | |
6. | Homework 6: observers and LQR | Dec 9 | solutions |
Exams
Important dates:
- Thu Sep 7: first lecture
- Thu Oct 5, Exam #1 (in class)
- Tue Nov 7, Exam #2 (in class)
- Thu Nov 23: Thanksgiving (no class)
- Tue Dec 12: Exam #3 (TBA: either in class or in the evening, also the last day of class)
- Other important dates (e.g. drop deadline): https://registrar.wisc.edu/fall_deadlines_at_a_glance.htm
Evaluation:
- Homework (25%): There will be roughly 6–8 homework assignments throughout the semester. You will have at least one week to complete each assignment. Each will be graded coarsely (e.g. out of 4 points). This is to give you a chance to apply the material we learn and to practice your skills. All assignments must be submitted through Gradescope. I will also post solutions to the class website afterward.
- Exams (75%): There will be three exams (25% each). The exams will be closed-book and closed-notes and will happen either during lecture or in the evening the same day as lecture. The exam dates are: October 5, November 7, and December 12.
Tentative syllabus:
The course is divided in three roughly equal parts and there is an exam after each part. Here is a tentative list of topics covered in each part.- Part I: state-space fundamentals. State-space models and modeling of physical systems, transfer functions, linearization, system responses, coordinate transformations, controllability and observability, canonical forms. Exam #1.
- Part II: analysis topics. Realization theory, minimal realizations, MIMO poles/zero cancellation, internal stability, BIBO stability, Lyapunov characterization. Exam #2.
- Part III: design and feedback. State feedback, pole placement, stabilizability/detectability, observer-based compensation, optimal control. Exam #3.
More information about the homework
- All homework assignments are due by 11:00pm on the date indicated.
- Assignments must be turned in electronically (as PDF or images) via Gradescope. For a tutorial on how to do this, please watch this video. If you’ll be scanning documents using a camera phone, read this guide.
- You may use a typesetting program (e.g. Word, LaTeX,…) or write neatly by hand. Either way, it must end up as a PDF or image files. If coding is required, include your MATLAB code as part of your solution. Printing to PDF or saving as PDF is usually the easiest way to go.
- It is your responsibility to ensure that what you turn in is legible, there are no pages missing, etc.
- Start each problem on a new page. If a problem has many parts, it’s OK to answer them on a single page.
- Explain your work. This means write in words how you solved the problem, and if the problem asks you to “show” something, your are expected to provide a mathematical proof. If code is required, use intuitive variable names, and comment any code you turn in.
- You are encouraged to discuss homework problems with classmates and even work in groups. However, the work you turn in must be your own. If you use any external sources (e.g. the internet) be sure to cite your sources!
Late policy:
- Homework assignments turned in late will not be accepted.
- It is your responsibility to ensure that you are available and present for the exams. The dates are already set, so plan accordingly!
- Exceptions will be made to the rules above in order to accommodate special circumstances. This includes family or medical emergencies, religious observances, and documented disabilities. If you have a special circumstance and foresee a conflict, please email the instructor as soon as possible to make alternative arrangements.
Discussion forum:
We will use Piazza (https://piazza.com/wisc/fall2017/ece717) for class-related discussions. Do not email the instructor directly with questions. Use Piazza or attend office hours!- On Piazza, anybody can ask questions, answer questions, or edit/improve existing responses. If you’re struggling with a concept, chances are you’re not alone—by posting your question on Piazza, it helps everybody!
- You also have the option to post/respond anonymously if you prefer.
- Do not use Piazza to ask for solutions, do not post solutions, and be nice to each other.
Textbook:
We will make use of the following textbook throughout the class.
- Robert L. Williams II and Douglas A. Lawrence. Linear State-Space Control Systems. John Wiley & Sons, Inc. 2007.
Several other textbooks can serve as auxiliary references. For linear algebra, I recommend:
- Gilbert Strang. Linear algebra and its applications.
- Sheldon Axler. Linear algebra done right.
- Panos J. Antsaklis and Anthony N. Michel. Linear systems.
- Chi-Tsong Chen. Linear system theory and design.