## CMPE 240 - Introduction to Linear Dynamical Systems

### Good Luck on the Final:

### Please use the Forum for Questions.

**Background**

*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 1^{st} order differential equations or 1^{st} order difference (or recursion) equations for discrete time systems. Generally, these systems can be written in a very simple (and very overloaded form) of:

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).

**Prerequisites**

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.

**Acknowledgements**

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

#### Handouts

- General Course Information
- Course Flyer
- A Primer on Matrices
- Basic Notation used in CMPE240
- Phase Plane mapping software in MATLAB and Java
- Least Squares in MATLAB
- Homework
- Homework Solutions
- Class Presentation Slides

#### 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.

- Lecture #0, 27-Sep-2007, Introduction to Linear Dynamical Systems.
- Lecture #1, 02-Oct-2007, Linear Functions and Applications.
- Lecture #2, 04-Oct-2007, Linear Algebra Review.
- Lecture #3, 09-Oct-2007, Linear Algebra Review (con't).
- Lecture #4, 11-Oct-2007, QR Decomposition
- Lecture #5, 16-Oct-2007, Least Squares
- Lecture #6, 18-Oct-2007, Least Squares Applications
- Lecture #7, 23-Oct-2007, Regularized Least Squares
- Lecture #8, 24-Oct-2007, Least Norm
- Lecture #9, 25-Oct-2007, Linear Dynamical Systems
- Lecture #10, 01-Nov-2007, Matrix Exponential
- Lecture #11, 06-Nov-2007, Eigenvalues and Eigenvectors
- Lecture #12, 11-Nov-2007, Diagonalization
- Lecture #13, 13-Nov-2007, Jordan Form
- Lecture #14, 15-Nov-2007, Linear Systems with Input and Output
- Lecture #15, 20-Nov-2007, Symmetric Matrices
- Lecture #16, 27-Nov-2007, Singular Value Decomposition
- Lecture #17, 29-Nov-2007, SVD Applications
- Lecture #18, 04-Dec-2007, Controllability
- Lecture #19, 06-Dec-2007, Observability
- Midterm Review, 01-Nov-2007, Midterm Review
- Final Review, 07-Dec-2007, Final Review
- Office Hours #2, 03-Oct-2007, Office Hours
- Office Hours #3, 08-Oct-2007, Office Hours
- Office Hours #4, 10-Oct-2007, Office Hours
- Office Hours #5, 15-Oct-2007, Office Hours
- Office Hours #6, 17-Oct-2007, Office Hours
- Office Hours #7, 22-Oct-2007, Office Hours
- Office Hours #8, 24-Oct-2007, Office Hours
- Office Hours #9, 19-Nov-2007, Office Hours
- Office Hours #10, 28-Nov-2007, Office Hours

#### Homework

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

licence to copy. The work you turn in should be your own.NOT

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

- Homework #1 Solution Set
- Homework #2 Solution Set
- Homework #3 Solution Set
- Homework #4 Solution Set
- Homework #5 Solution Set
- Homework #6 Solution Set
- Homework #7 Solution Set
- Homework #8 Solution Set
- Homework #9 Solution Set

#### Exams

- Practice Midterm (Solutions)
- Midterm: 01-Nov-2007, 24 Hour take-home exam, due 02-Nov-2007, Open Book, Notes, etc.
- Practice Final (Solutions)
- Final: 11 AM on 08-Dec-2007, 24 Hour take-home exam, due 11 AM on 09-Dec-2007, Open Book, Notes, etc.

#### 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 Presenterrather 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.

- Lecture #0: Introduction to Linear Dynamical Systems
- Lecture #1: Linear Functions
- Lecture #2: Linear Algebra Review
- Lecture #3: QR Factorization
- Lecture #4: Least Squares
- Lecture #5: Least Squares Applications
- Lecture #6: Regularized Least Squares
- Lecture #7: Least Norm
- Lecture #8: Autonomous LDS
- Lecture #9: Matrix Exponential
- Lecture #10: Eigenvectors
- Lecture #11: Jordan Form
- Lecture #12: Input Output
- Lecture #13: Symmetric Matrices
- Lecture #14: SVD Applications
- Lecture #15: Controllability
- Lecture #16: Observability
- Midterm Review (Part 1) 01-Nov-2007
- Midterm Review (Part 2) 01-Nov-2007
- Final Review 07-Dec-2007
- Office Hours 03-Oct-2007
- Office Hours 08-Oct-2007
- Office Hours 10-Oct-2007
- Office Hours 15-Oct-2007
- Office Hours 17-Oct-2007
- Office Hours 17-Oct-2007
- Office Hours 28-Nov-2007

#### General Class Information

Class Webforum:- WebForum - for announcements, general discussion, and help

Textbooks:note that these are NOT required, but are excellent referencesLinear Algebra and its Applications, 3rd Ed.by Gilbert Strang, Brooks Cole, 1988. ISBN: 0155510053.

- Name: Gabriel Hugh Elkaim (elkaim@soe.ucsc.edu)
- 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)