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Jack Baskin School of EngineeringUC Santa Cruz

CMPE 240 - Fall 2007

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CMPE 240 - Introduction to Linear Dynamical Systems

Good Luck on the Final:

Please use the Forum for Questions.


Linear Dynamical Systems (sometimes also called Linear Operator Theory refers to a mathematical representation of a physical system that can be represented by a set of 1st order differential equations or 1st order difference (or recursion) equations for discrete time systems. Generally, these systems can be written in a very simple (and very overloaded form) of:

xdot = A*x +B*u

The study of these linear systems started historically in the 1960's and required a Ph.D. in math as a necessary prerequisite. Most of the applications at the time were to aerospace control problems (such as rocket guidance). Today, these types of systems are studied extensively, and applications range from controls to economics. Frequently, these problems are cast as dual problems: design (where the input vector is altered to reach a desired output) and estimation (where a set a sensor measurements are processed to estimate the state of the system).


The only prerequisites for this class are exposure to Linear Algebra and Differential Equations (AMS/ENG 27 fulfills these just fine). A class on circuits (EE 70), controls (EE 154/241), signals and systems (EE 103), and/or dynamics (PHYS 5/6) would be useful, but are by no means critical. The only other prerequisites are a willingness to do the work, which will be hard at times.


This course is based on the Introduction to Linear Dynamical Systems sequence (EE263 and EE363), offered at Stanford by Professor Stephen Boyd. Lecture notes are taken from his published lecture notes, "EE263: Introduction to Linear Dynamical Systems," Fall 2004.

I would like to acknowledge the tremendous help and generosity of Prof. Stephen Boyd of Stanford University in teaching the subject matter to me, for all of his help with the slides, the homeworks, and the course materials. I would also like to thank Prof. Ed Carryer at Stanford University for pioneering this video capture technology, and helping me to set it up. Without their help and inspiration, this class would not be here.

Index of class resources

  • General Class Information — class and section times, instructor and TA information
  • Lecture Video — Video files of the lectures, and download information for the right codec.
  • Handouts — homework problem sets, homework solutions, other helpful handouts.
  • WebForum - for announcements, general discussion, and help


Lecture Videos

The technology to record these videos is supported by a grant from the Center for Teaching Excellence (CTE), and it is an experiment. Feedback as to the utility, and the usability of these videos would be highly appreciated. The basic hardware required is a tablet PC with the Office Tablet PC extensions, and a standard headset to capture the lecturers voice. Additionally, a program called Camtasia is used to capture the entire sequence into a standard movie format that can then be viewed at a later time for review and additional study.

You may view these lectures at any time, but do not distribute them beyond the UCSC environment. These lectures have been created using the Camtasia software, and can be played through the Camtasia player software, downloadable for free from techsmith here, or through the standard windows media player with the techsmith codec. A Mac OSX version of the codec can be found here that allows playback of the files. Note that some students have reported that VLC works much better on MacOSX and Linux.



Homeworks are handed out in class, and are due back either in class or in my office, 337B Engineering 2, at 6 PM on the following week. Homeworks will only be accepted at the beginning of class, not at the end of class. Homeworks turned in late will be receive half the total points once the solution set has been posted. Cooperation and collaboration on the homeworks is encouraged, but this is NOT licence to copy. The work you turn in should be your own.

  1. Homework #1: Introducation to Linear Dynamical Systems, Due 04-Oct-2007 (Solutions)
  2. Homework #2: Some Simple Design and Estimation, Due 11-Oct-2007 (Solutions)
  3. Homework #3: QR Factorization and Gram-Schmidt, Due 18-Oct-2007 (Solutions)
  4. Homework #4: Least Squares and Applications, Due 25-Oct-2007 (Solutions)
  5. Homework #5: Practice Midterm, Due 01-Nov-2007 (Solutions)
  6. Homework #6: Autonomous LDS and Matrix Exponential, Due 13-Nov-2007 (Solutions)
  7. Homework #7: Eigenvalues and Eigenvectors, Due 21-Nov-2007 (Solutions)
  8. Homework #8: Inputs and Outputs, Due 29-Nov-2007 (Solutions)
  9. Homework #9: SVD in all its glory, Due 06-Dec-2007 (Solutions)
  10. color_perception.m, required for homework #2.
  11. inductor_data.m, required for homework #3.
  12. deconv_data.m, required for homework #3.
  13. emissions_data.m, required for homework #4.
  14. beam_estim_data.m, required for homework #5.
  15. gate_sizing_data.m, required for homework #5.
  16. line_conv_data.m, required for homework #5.
  17. smooth_interpolation.m, required for homework #5.
  18. gauss_fit_data.m, required for homework #7.
  19. interconn.m, required for homework #8.
  20. time_comp_data.m, required for homework #8.
  21. mc_data.m, required for practice final.
  22. nleq_data.m, required for practice final.
  23. temp_prof_data.m, required for practice final.
  24., required for practice final.

Homework Solutions

The homework solutions are quite detailed, and are part of the required reading for the class. Note that by following what was the desired solution, you will see what we are trying to get you to learn with each specific homework problem.
  1. Homework #1 Solution Set
  2. Homework #2 Solution Set
  3. Homework #3 Solution Set
  4. Homework #4 Solution Set
  5. Homework #5 Solution Set
  6. Homework #6 Solution Set
  7. Homework #7 Solution Set
  8. Homework #8 Solution Set
  9. Homework #9 Solution Set


Class Presentation Slides

The class lectures use the digital ink capabilities of the TabletPC. The ink is saved back into the presentation, and the presentation is saved to the website for convenience. This year we are using Classroom Presenter rather than PowerPoint. It appears to be far more stable, and has several nice utilities for the TabletPC. The presentation files are in the .CSD format, and you will need to download Presenter to view them. Presenter can be downloaded free from here.

  1. Lecture #0: Introduction to Linear Dynamical Systems
  2. Lecture #1: Linear Functions
  3. Lecture #2: Linear Algebra Review
  4. Lecture #3: QR Factorization
  5. Lecture #4: Least Squares
  6. Lecture #5: Least Squares Applications
  7. Lecture #6: Regularized Least Squares
  8. Lecture #7: Least Norm
  9. Lecture #8: Autonomous LDS
  10. Lecture #9: Matrix Exponential
  11. Lecture #10: Eigenvectors
  12. Lecture #11: Jordan Form
  13. Lecture #12: Input Output
  14. Lecture #13: Symmetric Matrices
  15. Lecture #14: SVD Applications
  16. Lecture #15: Controllability
  17. Lecture #16: Observability

  19. Midterm Review (Part 1) 01-Nov-2007
  20. Midterm Review (Part 2) 01-Nov-2007
  21. Final Review 07-Dec-2007

  23. Office Hours 03-Oct-2007
  24. Office Hours 08-Oct-2007
  25. Office Hours 10-Oct-2007
  26. Office Hours 15-Oct-2007
  27. Office Hours 17-Oct-2007
  28. Office Hours 17-Oct-2007
  29. Office Hours 28-Nov-2007

General Class Information

Lecture times:
Tuesday-Thursday, 12:00 - 1:45 PM, Porter College Room #249
Class Webforum:
WebForum - for announcements, general discussion, and help
Textbooks: note that these are NOT required, but are excellent references

Linear Algebra and its Applications, 3rd Ed. by Gilbert Strang, Brooks Cole, 1988. ISBN: 0155510053.
Name: Gabriel Hugh Elkaim (
Phone: 831-459-3054
Office: Engineering 2, 337B
Instructor Office Hours:
Monday -- 2:00 - 4:00 PM, Wednesday -- 3:00 - 5:00 PM, and by appointment
Teaching Assistants:
TBD (unlikely to be any)