A beginning graduate course in multi-input multi-output, linear, time-invariant systems. Topics include state-space and input-output representations; solutions and their properties; multivariable poles and zeros; reachability, controllability, observability; stability; system norms and their computation; linearization techniques.
This is primarily a theory course aimed at students interested in dynamical systems with an emphasis on linear (or “linearized”) dynamics of input–output systems in the neighborhood of an equilibrium. This is not a modeling course per se, although some examples will include models of physical systems in engineering, biology, chemistry, etc. This course is well suited for beginning engineering graduate students (MS or PhD) or advanced undergraduate student interested in further education. This course is also of value of students in the physical or biological sciences interested in system dynamics modeling.
Audit registration is not permitted, but students are welcome to sit in on the course, please contact the instructor if you would like to sit in on the course.
3.0 Credit Hours
- Preliminaries: What is a linear dynamical system and how are they represented mathematically? Brief review of things you should already know: linear vector spaces, kernel, image, solution to linear algebraic equation, eigenvalues, eigenvectors, linearization, functions of a matrix, Cayley-Hamilton. (2)
- State equation representation, solutions to linear state equations for linear time-invariant (LTI) systems, transition matrix. Mostly focus on LTI, but mention fundamental results for linear time-varying systems (LTV). (4)
- Rugh Notes: Ch 1-3
- Rugh Book: Selected topics from Ch 2, 3, 5. (Skip 4)
- Bay: Ch 6
- Stability, including Lyapunov, BIBO (2)
- Rugh Notes: Ch 4
- Rugh Book: Selected topics from Cg 6, 7, 8.
- Bay: Ch 7
- Controllability and observability (2)
- Rugh Notes: Ch 5
- Rugh Book: Ch 9
- Bay: Ch 8
- Rugh NOtes: Ch 6
- State Feedback (2)
- Rugh Notes: Ch 8
- Rugh Book: Ch 14
- Bay: Ch 10
- Observers and Output Feedback (2)
- Rugh Notes: Ch 10
- Rugh Book: Ch 15
- Bay: Ch 10
- Stretch Goal: Geometric Theory: Invariant subspaces, stabilizability, noninteracting control (2)
- Rugh Book: Selected topics from Ch 18-19
Recommended Course Background: Undergraduate courses in linear algebra, differential equations, and an undergraduate level course in control systems. Students cannot take EN.530.616 if they have already taken a similar graduate-level Introduction to Linear Systems Theory course such as EN.520.601 OR EN.580.616.
Recommended course background: multivariable integral and differential calculus, classical physics, linear algebra, ordinary differential equations. Programming: Knowledge of the Matlab programming language including data input/output, 1-D and 2-D arrays, and user-defined function calls.
- Lectures: Weekly lectures in class and also available on the course Canvas site. You can stream them any time.
- Class Participation: Class participation represents approximately 10% of the course grade. If you are in a time zone which makes this difficult for you, please contact the instructor at the beginning of the course and we will accommodate you.
- Problem Sets: Problem sets based on the lectures and assigned reading are due at the on the date specified in the problem set. Problem sets represent approximately 45% of the course grade. All problem sets will be handed in on Gradescope.
- Quizzes given during class will likely represent approximately 10% of the course grade.
- Exams: Midterm and Final exams will represent approximately 45%.
- Each student’s lowest problem set score will be dropped.
- Problem Sessions: Live weekly online problem sessions will be held to answer questions regarding the lectures, problem sets, and exams.
- Any questions regarding grading must be referred to Prof. Whitcomb. The TA may not receive assignments or address grading questions.
Professor Louis L. Whitcomb
Department of Mechanical Engineering
G.W.C. Whiting School of Engineering
The Johns Hopkins University
Email: [email protected]
Office: 115 Hackerman Hall
Click HERE to email the instructor and TAs
When contacting the instructor and/or TAs please put “530.616” in the subject line of your email!
- Ms. Golchehr Amirkhani, [email protected],
- Mr. James Hunt, [email protected],
- Mr. Leo Mu, [email protected]
- Mr. Shuyuan Wang, [email protected]
If you have questions not answered on this page or the course Canvas site (registered students can access the Canvas site) please email the instructors and the TAs. Click HERE to email the instructor and TAs
Please include “530.616” in the subject line of all emails to the instructors regarding this course.
If You Have Questions Not Addressed On This Course Web Page
- 4:30PM-5:45PM ET Tuesdays and Thursdays in person on campus in Room 132 Gilman Hall. We will also stream the lectures online via ZOOM, and record the lectures for viewing on Canvas/Panopto. See announcement for details on the course Canvas site. If you are in a time zone which makes this difficult for you, please contact the instructor at the beginning of the course and we will accommodate you.
Problem Session and Office Hours
- 6:00PM-7:15PM ET Tuesdays in G50 Gilman Hall (basement level).
- 6:00PM-7:15PM ET Thursdays in 132 Gilman Hall (first-floor level).
- Midterm Exam: TBA
- Final Exam: 2PM–5PM Wednesday, December 13, 2023 per the Registrar’s Final Exam Schedule for Fall 2023.
- Required: W. J. Rugh, Notes on LTI Systems, Version 1.1, 2009. Available online here: https://sites.google.com/site/wilsonjrugh
- Strongly Recommended: W. J. Rugh (1996). Linear System Theory, 2nd Edition . Prentice-Hall. Every student of linear systems should own a copy of this book. This course used the second edition, not first edition. Highly useful chart of the book’s structure and chapter dependence is here: https://drive.google.com/file/d/1DA59b1y-NlIDb-g3rBSlOEOQ7RPDIUWD/view
- Recommended: J. S. Bay. Fundamentals of Linear State Space Systems. McGraw Hill. 1999. Available through open access: https://orb.binghamton.edu/electrical_fac/3
- Problem set assignments are available on Gradescope.
- All problem sets must be handed in on Blacboard. Most students will do problem sets by hand, and scan their assignment into a multi-page PDF to upload to Gradescope.
You may find it handy to have MATLAB available on your computer while reviewing the lectures and while doing the problem sets.
- Matlab is available on the computers in the Krieger Hall Computer Lab, Room 160 Krieger Hall, and also on the computers of the Robotics Teaching Lab, Room 170 Wyman Park Building.
- Johns Hopkins University has a Total Academic Headcount (TAH) site license for matlab. You can legally install matlab under the JHU site-license on your personal computers! Linux, Windows, and Mac-OS are supported. To get matlab for your PC:
- Browse to my.jhu.edu and login.
- Under the “Technology” glyph on the left, select the “mySoftware” menu item.
- Select the link “Software Catalog: To purchase new software”
- Search the software catalog for Matlab
- Follow the instructions on the page entitled “Matlab for Students”.
- On-line documentation for matlab is available on the PCs at the Krieger Academic Computing Laboratory. First run Matlab as described above, and type in the command “helpdesk<enter>”. Helpdesk will open in a web browser window.
- The “Getting Started in Matlab” tutorial introduction is available on-line from the mathworks web site: https://www.mathworks.com/help/pdf_doc/matlab/getstart.pdf
- Laboratory #0 asks you to read the “Getting Started in Matlab” tutorial from page 1 up to and including “scripts” and “functions”.
Notes and References
Supplementary material and links be posted online as needed.
- Gilbert Strang’s video lectures from his course 18.06 Linear Algebra are excellent and highly recommended: http://web.mit.edu/18.06/www/videos.shtml Some greatest hits include the following:
- Lecture #1: The Geometry of Linear Equations
- Lecture #6: Column Space and Nullspace
- Lecture #8: Solving Ax = b: Row Reduced Form R
- Lecture #9: Independence, Basis, and Dimension
- Lecture #10: The Four Fundamental Subspaces
- Lecture #21: Eigenvalues and Eigenvectors
Students are encouraged to work in groups to learn, brainstorm, and collaborate in learning how to solve problems.
Problem Sets and Lab Assignments Final Writeups
Your final writeups for pre-lab exercises and lab assignments must be done independently without reference to any notes from group sessions, the work of others, or other sources such as the internet.
While working on your final writeups for assignments, you may refer to your own class notes, your own laboratory notes, and the text.
Do Not Use Materials from Previous Years of This Course
You are NOT permitted to receive, give, or utilize any course materials from previous editions of this course. This ban includes all assignments, all exams, and all solutions. Use of these materials is an ethics violation that may result in punishment or dismissal from JHU.
Do Not USe Cheating Web Sites
Do not use cheating web sites like coursehero.com Uploading or downloading course materials to these cheating web sites is an ethics violation that may result in punishment or dismissal from JHU.
Disclosure of Outside Sources
If you use outside sources other than your class notes and your text to solve problems in the pre-lab and lab assignments (i.e. if you have used sources such as your roommate, study partner, the Internet, another textbook, a file from your office-mate’s files) then you must disclose the outside source and what you took from the source in your writeup.
In this course, we adopt the ethical guidelines articulated by Professor Lester Su for M.E. 530.101 Freshman experiences in mechanical engineering I, which are quoted with permission as follows:
Cheating is wrong. Cheating hurts our community by undermining academic integrity, creating mistrust, and fostering unfair competition. The university will punish cheaters with failure on an assignment, failure in a course, permanent transcript notation, suspension, and/or expulsion.
Offenses may be reported to medical, law or other professional or graduate schools when a cheater applies. Violations can include cheating on exams, plagiarism, reuse of assignments without permission, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition. Ignorance of these rules is not an excuse.
On every exam, you will sign the following pledge: “I agree to complete this exam without unauthorized assistance from any person, materials or device. [Signed and dated]”
For more information, see the guide on “Academic Ethics for Undergraduates” and the Ethics Board web site (http://ethics.jhu.edu).
I do want to make clear that I’m aware that the vast majority of students are honest, and the last thing I want to do is discourage students from working together. After all, working together on assignments is one of the most effective ways to learn, both through learning from and explaining things to others. The ethics rules are in place to ensure that the playing field is level for all students. The following examples will hopefully help explain the distinction between what constitutes acceptable cooperation and what is not allowable.
Student 1: Yo, I dunno how to do problem 2 on the homework, can you clue me in?
Student 2: Well, to be brief, I simply applied the **** principle
that is thoroughly explained in Chapter **** in the course text.
Student 1: Dude, thanks! (Goes off to work on problem.)
– This scenario describes an acceptable interaction.
There is nothing wrong with pointing someone in the right direction.
Student Y: The homework is due in fifteen minutes and I haven’t
done number 5 yet! Help me!
Student Z: Sure, but I don’t have time to explain it to you, so
here. Don’t just copy it, though.
(Hands over completed assignment.)
Student Y: I owe you one, man. (Goes off to copy number 5.)
– This scenario is a textbook ethics violation on the part of
both students. Student Y’s offense is obvious; student Z is
guilty by virtue of facilitating plagiarism, even though he/she
is unaware of what student Y actually did.
Joe Student: Geez, I am so swamped, I can’t possibly write up the
lab report and do the lab data calculations before it’s all due.
Jane student: Well, since we were lab partners and collected all
the data together…maybe you could just use my Excel spreadsheet
with the calculations, as long as you did the write-up yourself….
Joe Student: Yeah, that’s a great idea!
– That is not a great idea. By turning in a lab report with Jane’s
spreadsheet included, Joe is submitting something that isn’t his
Study group member I: All right, since there’s three of us and
there’s six problems on the homework, let’s each do two. I’ll
do one and two and give you copies when I’m done.
Study group member II: Good idea, that’ll save us a lot of work.
I’ll take three and five.
Study group member III: Then I guess I’ll do four and six. Are you
guys sure this is OK? Seems fishy to me.
Study group member I: What’s the problem? It’s not like we’re
copying the entire assignment. Two problems each is still a lot
– This is clearly wrong. Copying is copying even if it’s only
part of an assignment.
Mike (just before class): Hey, can you help me? I lost my
calculator, so I’ve got all the problems worked out but I
couldn’t get the numerical answers. What is the answer for
Ike: Let’s see (flips through assignment)… I got 2.16542.
Mike: (Writing) Two point one six five four two…what about
Ike: For that one… I got 16.0.
Mike: (Writing) Sixteen point oh…great, got it, thanks.
Helping out a friend totally rules!
– Helping out a friend this way does not rule, totally or
partially. As minor as this offense seems, Mike is still
submitting Ike’s work as his own when Mike gets the numerical
answer and copies it in this way.