## Ee263 Homework Solutions Boyd

### Good Luck on the Final

### Use the forum for final 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), 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**

**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.

- Lecture #0, 22-Sept-2005, Introduction to Linear Dynamical Systems.
- Lecture #1, 27-Sept-2005, Linear Functions.
- Lecture #2, 29-Sept-2005, Linear Algebra Review.
- Lecture #3, 05-Oct-2005, Linear Algebra Review (con't).
- Lecture #4, 06-Oct-2005, Orthonormal Vectors and QR Factorization
- Lecture #5, 11-Oct-2005, Least Squares Approximate Solution
- Lecture #6, 14-Oct-2005, Least Squares Applications
- Lecture #7, 18-Oct-2005, Multi-Objective Least Squares
- Lecture #8, 20-Oct-2005, Min-Norm Solutions
- Lecture #9, 25-Oct-2005, Autonomous Linear Dynamical Systems
- Lecture #10, 27-Oct-2005, Laplace Transforms and State Transition Matrices
- Lecture #11, 1-Nov-2005, The Matrix Exponential
- Lecture #12, 3-Nov-2005, Eigenvalues and Eigenvectors
- Lecture #13, 8-Nov-2005, Jordan Canonical Form
- Lecture #14, 10-Nov-2005, Cayley-Hamilton and LDS with I/O
- Lecture #15, 15-Nov-2005, Impulse and Step Response matrices
- Lecture #16, 17-Nov-2005, Symmetric Matrices and Quadratic Form
- Lecture #17, 22-Nov-2005, Singular Value Decomposition
- Lecture #18, 29-Nov-2005, Controllability and Reachability
- Lecture #19, 1-Dec-2005, Observability

**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 Tuesday of 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-2005
- Homework #2: Some Simple Design and Estimation, Due 11-Oct-2005
- Homework #3: QR Factorization and Gram-Schmidt, Due 18-Oct-2005
- Homework #4: Least Squares and Least Norm Solutions, Due 25-Oct-2005
- Homework #5: Autonomous Linear Dynamical Systems, Due 03-Nov-2005
- Homework #6: Linear Systems and the Matrix Exponential, Due 15-Nov-2005
- Homework #7: Linear Systems with Inputs and Outputs, Due 23-Nov-2005
- Homework #8: Quadratic forms, SVD, Due 29-Nov-2005
- color_perception.m: MATLAB file for Homework #2
- inductor_data.m: MATLAB file for Homework #3
- deconv_data.m: MATLAB file for Homework #3
- emissions_data.m: MATLAB file for Homework #4
- interconn.m: MATLAB file for Homework #7
- time_comp_data.m: MATLAB file for Homework #7

**Exams**

**Homework Solutions**

- 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

**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 apprears 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 here.

- Lecture #0: Introduction to Linear Dynamical Systems
- Lecture #1: Linear Functions
- Lecture #1(b): Example Problems
- Lecture #2: Linear Algebra Review
- Lecture #3: Orthonormal Basis and QR decomposition
- Lecture #4: Least Squares
- Lecture #5: Least Squares Applications
- Lecture #6: Regularized Least Squares
- Lecture #7: Least Norm and Minimum Norm Solutions
- Lecture #8: Autonomous Linear Dynamical Systems
- Lecture #9: The Matrix Exponential
- Lecture #10: Eigenvalues and Eigenvectors
- Lecture #11: The Jordan Form
- Lecture #12: Linear Systems with Inputs and Outputs
- Lecture #13: Symmetric Matrices, Quadratic Forms, SVD
- Lecture #14: Singular Value Decomposition Applications
- Lecture #15: Controllability
- Lecture #16: Observability

**General Class Information**

Lecture times:- Tuesday-Thursday, 10:00 - 11:45 PM, E2-506

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

Textbooks:note that these are NOT required, but are excellent referencesLinear System Theory and Designby Chi-Tsong Chen, Oxford University Press, 1999. ISBN: 0030602890.Linear Systemsby Thomas Kailath, Prentice-Hall, 1980. ISBN: 0135369614.Linear Algebra and its Applications, 3rd Ed.by Gilbert Strang, Brooks Cole, 1988. ISBN: 0155510053.

Instructor:

- Name: Gabriel Hugh Elkaim (elkaim@soe.ucsc.edu)
- Phone: 831-459-3054
- Office: Engineering 2, 337B

Instructor Office Hours:- Tuesday-Thursday, 2:00 - 4:00 PM, and by appointment

Teaching Assistants:- TBD (unlikely to be any)

## Instructor

Reza Nasiri Mahalati

Office hours: Thursday 10:30AM - noon (after lecture) in Mitch B67

## Lectures

06/26/2017 - 12/7/2017

Tuesdays and Thursdays, 9:00AM - 10:20AM in NVIDIA Auditorium

No lecture on 11/21 or 11/23 for Thanksgiving

## Review sessions

Weekly review sessions will be held on Fridays (starting 9/29)

From 1:30PM to 2:20PM in NVIDIA Auditorium

These sessions will be videotaped by SCPD and uploaded on their website the following week. Notes from the review sessions will be posted under the notes tab of the website.

## Exams

The midterm exam will be a

**12hr take-home**. Students can choose which date and time to take the midterm. Submissions are due on gradescope 12hrs after your start.11/3: 5:00-5:30PM

11/4: 10:00-10:30AM or 5:00-5:30PM

11/5: 10:00-10:30AM or 5:00-5:30PM

**IMPORTANT**: The midterm exam distribution will be online over email. All of the students are required to go to this link, and sign up for one of the 5 exam times by 11/2 at noon.

The final exam will be a

**15hr take-home**. Students can choose which date and time to take the final. Submissions are due on gradescope 15hrs after you start.12/8: 5:00-5:30PM

12/9: 9:00-9:30AM or 5:00-5:30PM

12/10: 9:00-9:30AM or 5:00-5:30PM

**IMPORTANT**: The final exam distribution will be online over email. All of the students are required to go to this link, and sign up for one of the 5 exam times by 12/6 at noon.

## Course description

Applied linear algebra and linear dynamical systems with applications to circuits, signal processing, communications, and control systems. Topics: least-squares approximations of over-determined equations, and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm, and singular-value decomposition. Eigenvalues, left and right eigenvectors, with dynamical interpretation. Matrix exponential, stability, and asymptotic behavior. Multi-input/multi-output systems, impulse and step matrices; convolution and transfer-matrix descriptions. Control, reachability, and state transfer; observability and least-squares state estimation.

Prerequisites: linear algebra and matrices as in MATH104; differential equations and Laplace transforms as in EE102A.

## Textbooks

There are no required or optional textbooks. Complete notes will be available online. See the section on reading for details.

## Announcements

We are using Piazza. We'll post all announcements there, not here, so make sure you join.

## Students with Documented Disabilities

Ensuring that students with disabilities have full access in all instructional settings is one of our highest priorities. Students who may need an academic accommodation based on the impact of a disability must initiate the request with the Office of Accessible Education (OAE). Professional staff will evaluate the request with required documentation, recommend reasonable accommodations, and prepare an Accommodation Letter for faculty. Unless the student has a temporary disability, Accommodation letters are issued for the entire academic year. Students should contact the OAE as soon as possible since timely notice is needed to coordinate accommodations. The OAE is located at 563 Salvatierra Walk (phone: 723-1066).

## Archive

This course was originally developed and taught by Professor Stephen Boyd, and the complete set of materials consisting of lecture videos, slides, support notes and homework is still available in the archive.

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